Mechanical characterization and simulation of fracture processes in polysilicon Micro Electro

Mechanical Systems (MEMS)

Tesi da presentare per ilconseguimento del titolo di Dottore di Ricerca

Politecnico di MilanoDipartimento di Ingegneria Strutturale

Dottorato in Ingegneria Strutturale, Sismica e Geotecnica - XIX Ciclo

diFabrizio Cacchione

Aprile 2007

Mechanical characterization and simulation of fracture processes in polysilicon Micro Electro Mechanical Systems (MEMS)

Tesi di Dottorato dell’ Ing. Fabrizio CacchioneRelatore:Prof. Ing. Alberto Corigliano

Aprile 2007

Dottorato in Ingegneria Strutturale, Sismica e Geotecnica del Politecnico di Milano

Collegio dei Docenti:Pietro Gambarova (Coordinatore)Annamaria CividiniClaudia ComiRoberto ControAlberto CoriglianoClaudio di PriscoMarco di PriscoAlberto FranchiCarmelo GentileCristina JommiGiulio MaierPaolo NegroRoberto NovaAnna PandolfiRoberto PaolucciMaria Adelaide ParisiUmberto PeregoFederico PerottiAlberto Taliercio

Acknowledgments

The first acknowledgment goes to Dr. Benedetto Vigna and to all the persons in the Mems Product Division of STMicroelectronics who believed in me and gave me the opportunity to continue my research in the very interesting and stimulating field of microsystems, by sponsoring the PhD grant. I desire to express my gratitude to my supervisor Prof. Alberto Corigliano for his scientific guide and for his aid along these three years. During this thesis I had the opportunity to interact with many researchers of the scientific board at the Department of Structural Engineering of the Milan Polytechnic: I would like thank to them all for the suggestions and the kind support. I would like to express my gratefulness to: Prof. Attilio Frangi to have introduced me to Fortran programming, for the very helpful suggestions he gave me in many occasions and for transmitting me every time enthusiasm and a 'positive-thinking' way to face everyday problems; Dr. Aldo Ghisi, for the support given with free fall simulations, but mostly for his capacity to listen to my problems and to give me right indications and suggestions; Dr. Stefano Mariani, a very precious person, with lot of new and interesting ideas and with a personality I really appreciate; Prof. Anna Pandolfi, who helped me in the solution of some problems when I did not know how to get out.Many thanks to all the guys in the Department I met by the way, I spent some nice time with in different occasions, and shared a piece of my life with.Finally, I wish to thank for the constant love and comprehension the persons who always make my dreams possible, to whom this work is dedicated.

To Elisa and to my family.

Milan, April 2007Fabrizio Cacchione

Contents

1 Introduction.................................................................................................... 71.1.Engineering motivations........................................................................ 71.2.Objectives, methodology and outline of the thesis............................ 8

1.2.1.Methodology.................................................................................... 81.2.2.Thesis layout.................................................................................... 9

2 Micro Electro Mechanical Systems (MEMS)........................................... 112.1.General description............................................................................... 112.2.Design of a suspended structure with a simple micromachining process.......................................................................................................... 122.3.Applications........................................................................................... 13

2.3.1.Pressure sensors............................................................................ 132.3.2.Accelerometers.............................................................................. 152.3.3.Gyroscopes..................................................................................... 172.3.4.RF switches.................................................................................... 19

3 Mechanical characterization of polysilicon as a structural material for MEMS................................................................................................................ 21

3.1.Polysilicon as a structural material in MEMS.................................... 213.2.Testing methodologies.......................................................................... 233.3.Quasi-static testing................................................................................ 25

3.3.1.Off chip tension test...................................................................... 263.3.2.Off chip and on-chip bending test............................................... 273.3.3.Test on membranes (Bulge test)................................................... 293.3.4.Nanoindenter-driven test............................................................. 30

3.4.High frequency testing......................................................................... 323.4.1.Fatigue testing and on-chip structures....................................... 333.4.2.Fatigue mechanisms...................................................................... 33

4 Weibull theory applied to the study of polysilicon strength................364.1.General concepts................................................................................... 364.2.Basic theory............................................................................................ 374.3.Statistical size effect and stress gradient effect.................................. 39

4.3.1.Effect of the modulus.................................................................... 404.3.2.Effect of the volume...................................................................... 414.3.3.Effect of the stress distribution.................................................... 42

4.4.Application of Weibull approach to 3D structures........................... 464.4.1.Failure probability computation.................................................. 46

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4.4.2.Weibull parameters identification............................................... 475 On-chip testing of ThELMATM polysilicons............................................ 49

5.1.The ThELMATM process........................................................................ 495.2.Study of the role of grain structure and surface defects in thin polysilicon.................................................................................................... 52

5.2.1.Out of plane structure description.............................................. 525.2.2.Numerical models for the computation of electrostatic fields.545.2.3.Data reduction procedure............................................................ 575.2.4.Results and discussion.................................................................. 61

5.3.Design and testing of a new test structure for the mechanical characterization of thick polysilicon......................................................... 63

5.3.1.Structure description..................................................................... 635.3.2.Data reduction procedure............................................................ 685.3.3.Result and discussion................................................................... 695.3.4.Estimation of fracture parameters............................................... 73

6 Numerical representation of polycrystals................................................ 866.1.Materials and microstructures............................................................. 866.2.Voronoi tessellation as a tool for the creation of numeric polycrystals.................................................................................................. 88

6.2.1.Definition and properties............................................................. 896.2.2.Centroidal Voronoi tessellation................................................... 916.2.3.Creation of interphases................................................................. 936.2.4.MEMS structures tessellation....................................................... 94

6.3.Constitutive models for polysilicon.................................................... 966.3.1.Elastic tensor for cubic crystals.................................................... 966.3.2.Non linear cohesive crack model.............................................. 101

7 Non linear mechanical simulation algorithms for MEMS.................. 1077.1.Overview on MEMS mechanical simulations.................................. 1077.2.Direct step by step dynamic analysis................................................ 108

7.2.1.Explicit Methods.......................................................................... 1097.2.2.Implicit Methods......................................................................... 111

7.3.Dynamic relaxation algorithm for quasi-static simulations........... 1127.4.Cohesive fracture algorithm.............................................................. 1167.5.Pure explicit algorithms for micro-scale simulations: drawbacks.1197.6.Implicit–explicit integration scheme................................................. 1217.7.FETI algorithms for multi-domain problems................................... 123

7.7.1.Comberscure-Gravouil algorithm............................................. 123

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7.7.2.Application of the domain decomposition algorithm to multi-connected domains with crack propagation..................................... 128

7.8.Mass scaling and final remarks......................................................... 1368 Parametric study of polycrystalline silicon............................................ 139

8.1.Methodology adopted........................................................................ 1398.2.Young's modulus evaluation and prediction.................................. 139

8.2.1.Grain size effect........................................................................... 1408.1.1.Texture effect............................................................................... 142

8.3.Fracture properties.............................................................................. 1448.3.1.Grain orientation effect............................................................... 1468.3.2.Specimen size effect.................................................................... 1478.1.2.Grain size effect........................................................................... 1498.3.3.Introduction of defects................................................................ 152

9 Fracture and shock assessment of MEMS accelerometers................... 1589.1.Methodology adopted........................................................................ 1589.2.Shock assessment of an uniaxial accelerometer.............................. 159

9.2.1.Global level.................................................................................. 1599.2.2.Device level.................................................................................. 1619.2.3.Local level..................................................................................... 163

10 Conclusions............................................................................................... 16610.1.Achieved results................................................................................ 166

10.1.1. Experimental results................................................................ 16610.1.2. Numerical procedures and algorithms.................................. 16710.1.3. Parametric simulations............................................................ 16810.1.4. Simplified approach for the shock assessment..................... 168

10.2.Future prospects................................................................................ 16810.2.1. Fatigue and fracture testing.................................................... 16810.2.2. Numerical models.................................................................... 169

References....................................................................................................... 170

III

Introduction

1.1. Engineering motivations

The diffusion of micro-electro-mechanical-systems (MEMS) devices in a wide range of applications is rapidly increasing over the years. This depends on the relatively low cost of the applications produced with micromachining technologies with respect to equivalent products fabricated using standard technologies.One of the key factors on the final cost of the products is the size of the device: scaling down the size of the products makes possible an increase of the number of devices that can be produced with the same amount of material and with the same fabrication time and therefore it lowers the price per produced unit.In the same time, from a structural engineering point of view, the design of a micro-structure made of very tiny beams and plates arises a number of questions. It is not completely clear if at this scale it is still possible to use the same methodologies, models and tools commonly used at the macro-scale. In the case of a positive answer, it should be interesting to know the limit, intended as a characteristic structural dimension, for the application of classical structural engineering tools (like structural beam and plate theories). Among many, some open questions at the micro-scale are: if the material can be modeled as homogeneous and isotropic; how the microstructural morphology plays its role in the determination of mechanical properties of the material; if the material mechanical behavior would be the same at the micro-scale and at the macro-scale and how would it be possible a direct measure of the mechanical properties of the material.From an academical point of view, these questions opened the way to new research fields that can be seen as a meeting point for the classical structural engineering, material science and engineering, physics and chemistry.

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1.2. Objectives, methodology and outline of the thesis

The aim of this thesis is to investigate and to give an answer to some of the questions formulated above, combining both experimental and numerical techniques to study the mechanical behavior of the most common structural material used for microsystem applications, polysilicon. The main objectives are:

• the investigation of elastic and fracture properties of the material;• the critical evaluation of the fracture assessment techniques used

by the scientific and industrial community;• the possibility to accurately simulate the mechanical behavior of

the material in order to use fast an reliable numerical tools instead of very time and money consuming experimental campaigns to understand the role of the microstructure on mechanical properties.

• Since impacts and shock waves are one of the principal causes of failure for MEMS devices, algorithms and models were studied in order to simulate the mechanical response of the structure and a simplified approach for the shock assessment was proposed.

1.2.1. MethodologyThe experimental work done in this thesis is accomplished using diverse techniques: optical and electron scanning microscopy were used to measure and control the layout of the structure designed, the fracture paths and the operating conditions of the devices; electrical measurements were performed using high resolution electronic instrumentation and electrical schemes quite common for mechanical characterization at the microscale, such as voltage generators to apply the structural load and capacimeters to measure the displacement of the specimens.Numerical analyses were performed using both three-dimensional and bi-dimensional models. The Ansys Inc. software was used in the microstructure design phase and for linear elastic static simulations.

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Abaqus program was instead used for explicit dynamics simulations performed for the shock assessment and discussed in Chapter 9. A non -linear static and implicit-explicit dynamic code was instead developed and used to carry out parametric fracture simulations and dynamic fracture simulations. This code includes the traditional displacement finite element formulation enriched by other advanced tools, such as: the microstructural description of the material; the run-time insertion of interface elements in order to simulate the fracture propagation and a multi-domain integration algorithm used to bypass some computational difficulties connected with explicit integration algorithms.

1.2.2. Thesis layoutAfter this introduction, in Chapter 2 a short introduction to Micro Electro Mechanical Systems (MEMS) is presented. Some commercially available devices are described, together with an example of a simple deposition process invented to give an idea on how the fabrication of a suspended microstructure is possible.In Chapter 3 the state of the art on the mechanical characterization of polysilicon is given and the most important experimental techniques used are described. The major advantages of each technique are underlined as well as the results achieved and some critical consideration about the progresses that should be done to have a more accurate mechanical material characterization.The Weibull approach, commonly used to characterize fracture properties of brittle materials is presented in Chapter 4. The capability of this method in foreseeing the fracture loads of micromachined components in complex stress conditions is carefully analyzed and a numerical procedure for the reduction of Weibull parameters from an experimental data set is described.All the experimental work done is treated in Chapter 5. Two microstructures, designed and subsequently tested are described and the results obtained by the use of the on-chip approach are discussed. In the final part of the Chapter, some hypotheses for a fracture mechanics material characterization are considered.

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Chapter 6 is focused on the numerical representation of polycrystals by the use of routines that are capable to divide a domain into tassells, whose shape looks like a typical granular structure of a polycrystal. Besides, the constitutive models adopted to describe material behavior are presented.Numerical algorithms used in the microstructural analyses are addressed in Chapter 7. The main problems and peculiarities regarding the solution of the dynamic problem at the micro-scale are examined and the solutions adopted for the reduction of the computational cost of the simulations are tackled.The influence of the grain size on the elastic properties of the material, as well as the role of a privileged grain orientation is investigated in the first part of Chapter 8. In the second part of the chapter the role of the microstructural morphology, the effect of the size of the structure and the one of a possible defect distribution on the overall fracture properties are analyzed.A possible simplified approach for the mechanical shock assessment of a MEMS accelerometer is given in Chapter 9. The problem is divided into three simpler sub-problems with a computational cost by far smaller than the one of the initial problem solved monolithically.Work for the future and research perspectives are confined to the final Chapter 10, where also the conclusions of this work are summarized.

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Chapter 2

Micro Electro Mechanical Systems (MEMS)

2.1. General description

Over the years conventional microelectronics has continued to make spectacular advances, following Moore’s law to deliver undreamed processor speeds, integration density and memory capacity. However, this phenomenon has been accompanied by an increase in specialization where most engineers focus on some specific aspect of a circuit design and refine it to a very fine art.More recently, however, a new application of integrated circuit fabrication technology has emerged that has brought a significantly new approach to design and development: Micro-electro-mechanical systems, or MEMS. These devices are constructed using fabrication techniques familiar to the semiconductor industry, manufacturing components in large batches on silicon wafers. The difference is that they combine with electrical and electronic components also mechanical, optical or fluidic elements. MEMS extend the functionality of silicon components into many new applications such as accelerometers or laboratory-on-chip products. A key benefit of this technology is that it builds on well known manufacturing techniques, allowing also the use of older equipment since the lithography is not deep submicron. Moreover, since the components are made side-by-side on wafers and with an extremely well controlled process they can be much more precise and repeatable than similar products manufactured in other ways.

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Microelectronic integrated circuits (ICs) can be thought of as the "brains" of systems and MEMS augments this decision-making capability with "eyes" and "arms", to allow microsystems to sense and control the environment. In its most basic form, the sensors gather information from the environment through measuring mechanical, thermal, biological, chemical, optical, and magnetic phenomena; the electronics process the information derived from the sensors and through some decision making capability direct the actuators to respond by moving, positioning, regulating, pumping, and filtering, thereby controlling the environment for some desired outcome or purpose.Today the most common MEMS devices are mechanical sensors that measure acceleration, rotation and pressure. On the other side, the most common MEMS used not for sensing some external mechanical stimulus, but to act on a system is the inkjet thermal printer head chip, that represents the most known product in a very fast expanding technology: microfluidic systems. Hundreds of microscopic channels are created inside this chip and are normally filled with ink. Small resistive heating elements in each channel can be independently driven so that the temperature in a limited area rises to around 800°C in a few microseconds, vaporizing part of the ink to propel the ink volume out of the channel towards the paper, painting a small dot.

2.2. Design of a suspended structure with a simple micromachining process

In this section an example will be given on how a micromachining process can be used for the design of a suspended structure. For this purpose, a surface micromachining process is invented. This process was thought to be as simple as possible, in order to give to the reader an idea of a MEMS fabrication process. The result of the process is a cantilever beam clamped to the substrate. It is worth noting that all of the steps described are just the sum of many technological processes, that have the aim to create an eigen-stress free material, with a very precise thickness and dimensional control, without

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defects and with desired physical-chemical properties.With the aid of Figure 2.1 it is possible to describe the process. Starting from the monosilicon substrate (Figure 2.1(1)) it is possible to grow a film of silicon oxide (Figure 2.1(2)). This layer is needed to create the gap between the movable structure and the substrate, as will be clarified later, and it is not strictly necessary for the construction of structural parts of the MEMS. The subsequent step consists in creating a square hole into the silicon oxide layer (Figure 2.1(3)). The polysilicon film is then deposited on the oxide (Figure 2.1(4)). The cavity made into the oxide is thus filled with polysilicon. In the place of the hole is therefore created the mechanical connection between the polysilicon and the substrate. An etching process is then needed to give the desired shape to the mechanical structure (Figure 2.1(5)). At this point the structure is not free to move, because it is glued to the substrate by the oxide layer. The last step consists into the removal (usually with a dry acid attach) of the oxide (Figure 2.1(6)).

2.3. Applications

This section gives an introduction to some MEMS sensors currently in use and explains their principles of operation. The section covers pressure sensors, accelerometers, gyroscopes and RF switches, that (excluding microfluidics MEMS), are the most common micromachined sensors.

2.3.1. Pressure sensorsPressure sensors are one of the most commonly used forms of MEMS sensors. They are found in a wide and expanding area of applications ranging from blood pressure monitoring, washing machines, car tires and exhausts to hydraulic systems and aeronautics.Pressure sensors can be classified into two main families, distinguishing the way they sense pressure.

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Figure 2.1. Simple surface micromachining process. (1) Silicon substrate. (2) Oxide growth. (3) Oxide etching. (4) Deposition of the structural layer. (5) Structural

layer etching. (6) Oxide removal.

The most simple and common ones are the piezoresistive devices. They are principally thin plates with the external boundary clamped to the substrate, as shown in Figure 2.2(a). When the pressure acts on the top surface of the device, causes the plate to deform. Nearby the edges of the membrane, where the strains are maximum, are usually placed four piezo-resistors. Thus, when the plate undergoes bending, the resistance of the piezo-resistors changes. By measuring the resistance changes of the system, it is possible to know the value of the pressure acting on the diaphragm.Another common layout of pressure sensors relies on a different working principle. As it is possible to see in Figure 2.2(b) , the entire system acts as a capacitor. In fact, when the pressure deforms the plate, it moves toward an electrode placed on the bottom. Diminishing the distance between the plate and the electrode causes the total capacitance to increase.

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Figure 2.2. (a) Side and top views of a piezoresistive pressure sensor (from Johnson, 1992); (b) side view of a capacitive pressure sensor (from Lee and Wise,1982)

Therefore, knowing the relationship between the deformed configuration of the plate versus the capacitance variation and the one between the applied pressure versus the deformed shape of the plate, it is possible to measure the pressure starting from the capacitance changes.

2.3.2. AccelerometersAccelerometers or inertial sensors are widely used within the aerospace, defence, automotive, marine and consumer industries. In the aerospace industry they are used for flight stabilization of aircraft and rockets and navigation. Automotive applications include vehicle stability systems, rollover prevention systems and aids to navigation as well as impact sensors in airbags. Naval and marine applications include ship stabilization and navigation. The most common MEMS accelerometer designs fall into two categories: capacitive and piezoresistive.Figure 2.3 illustrates the working principle of a very simple capacitive accelerometer. The device can be schematized as a second order mechanical system, composed by a mass, a spring and a damper.

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Figure 2.3. Schematic of a capacitive accelerometer

When the external acceleration acts on the system, the mass moves in the opposite direction of the acceleration. The displacement is then sensed by a capacitive system. It is formed by a number of electrodes that are connected with the mass and therefore are free to move and others that are clamped to the substrate. The displacement of the mass causes a change in the capacitance measured by the electrostatic system. Thus from the direct measure of this change is possible to compute the value of the external acceleration.The mechanisms of a piezoresistive accelerometer is very similar to the one of the piezoresistive pressure sensor. Some piezoresistors (in black in Figure 2.4) are diffused on the top surface of the device and electrically connected in order to form a Wheatstone bridge. When the acceleration acts on the system, the proof mass reacts moving and deforming the thin plate it is suspended above (as schematically shown in Figure 2.4). The deformation of the plate causes the resistance change of the piezoresistors. The electronic circuit senses the resistance change and computes the magnitude and direction of the acceleration from the electric signal. MEMS accelerometer performance relies on many of the same parameters as MEMS pressure sensors. The primary metrology issues are accurate thickness control and accurate dimensional manufacturing of the proof mass.

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Figure 2.4. Schematic of a piezoresistive accelerometer

Of relevant importance are the material properties, such as Young’s modulus, which affect device sensitivity, particularly in suspension beams. Sidewall orthogonality and parallelism is also an issue for accelerometers as deviations from parallel sidewalls can lead to device non-linearity and failure.

2.3.3. GyroscopesAs with accelerometers, gyroscopes are increasingly being used in consumer products. In parallel to this, applications in the traditional gyroscope markets, such as aeronautics and defense, are expanding. Macro-scale gyroscopes normally use a large mass flywheel rotating at high speed, however the entity of frictional forces, that cause a premature failure of the system and lubrication techniques, not sufficiently efficient at this scale, prevent this technique being viable for microdevices. As a result, most micromachined gyroscopes use a mechanical structure that is

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driven into resonance and rotation excites a second resonance due to the Coriolis force (see Figure 2.5). Considering the mechanical scheme of the gyroscope, it is possible to write the equations of motion in the local frame. They are:

{m xx xk x x=F xm x 2m y 2m y m y y yk y y=F ym y 2−m x −2m x

, (2.1,2.2)

where m is the mass of the system, x and y the local reference coordinates, x and y the damping coefficients, k x and k y the elastic stiffnesses, F x and F y the forces acting on the system and is the rotation of the local frame with respect to an inertial system.

Figure 2.5. Schematic of a MEMS gyroscope

In order to measure the angular rate with this gyroscope, the x mode is driven sinusoidally using F x at an amplitude of Fd and a pulsation of d. The pulsation d is usually significantly higher than the specified bandwidth of the gyroscope. At this high frequency, the terms x 2, y 2, y and x are small and can often be neglected. The equations of motion become:

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{m xx xk x x=F x2m y m y y yk y y=F y−2m x

, (2.3,2.4)

When the gyroscope experiences an external angular velocity, =, the 2 x term in equation (2.4) causes the y mode to vibrate at the driven frequency d with an amplitude that is proportional to the angular rate, . Therefore from the y output signal, the value of the angular rate can be obtained.An obstacle to this technology is the small size of the Coriolis force compared to the driving force that is applied in an orthogonal direction. One way of countering this is using structures with a high output over input ratio, as for instance structures vibrating at the resonance of the sensing axis. Unsatisfactory alignment of the drive mechanism with the axis of freedom can also result in cross-talk, which can overshadow the relatively small Coriolis force.MEMS gyroscope performance relies on many parameters including the dimensional and material properties of the resonators and the suspension beams. Resonant frequencies of the gyroscopes are often measured with vibrometers, as extra modes can cross-talk with Coriolis induced modes. Suspension beams require dimensional measurements including sidewall verticality and thickness measurement.

2.3.4. RF switchesHigh frequency switches are used in wireless communications and radar systems for switching between the transmit and receive paths, for routing signals to the different blocks in multi-band/standard telephones, for RF signal routing in phase shifters used in phased-array antennas, and numerous other applications. RF-MEMS switches offer significant benefits over semiconductor switches in terms of high isolation (in particular over 30 GHz), low loss over a wide frequency range, extremely low standby power consumption and excellent linearity characteristics. However, the main drawbacks remain the relatively high drive voltages and slow response.

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Figure 2.6 shows a very common RF switch layout. When the line that brings the signal is at the same potential of the bridge, no forces act on the system and therefore the switch is in the up state. If a constant voltage is applied on the signal line, an electrostatic force arises that pushes the bridge toward the line. If the voltage is bigger than a threshold value, the bridge collapses on the line causing a change in the state of the switch, that goes in the down position.

Figure 2.6. Side and top views of a RF switch (from Tilman, 2003)

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Chapter 3

Mechanical characterization of polysilicon as a structural material for MEMS

3.1. Polysilicon as a structural material in MEMS

Polysilicon is by far the most common structural material in MEMS applications. It is used for a huge variety of applications, mainly because of two factors:

• the existence of well-established deposition technologies, in which polycrystalline silicon had a very important role since the beginning of the microelectronics era;

• the excellent physical properties of this material. Its Young’s modulus is higher than that of titanium and comparable with that of steel. The rupture resistance at the micro-scale stands in the range of the one of the best construction steels, while its density is less than aluminum's one. Thermal properties, too, make it a very good material for high temperature applications. Thermal conductivity is very high, while the thermal expansion coefficient is very small and the melting point is only one hundred degrees less than iron’s one.

Since polysilicon is an aggregate of mono-crystalline silicon grains, its properties depend on the properties of the crystallites composing it, on their shapes, on their orientation and on the physical characteristics of the

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grain boundaries between different grains. This means that the overall properties of polysilicon will be strongly influenced by the process used for the deposition; process data like the deposition temperature and the deposition pressure will all influence the final properties of the material. Besides the dispersion of the physical properties due to different processes, it has to be added that at the micro-scale the measure of these properties is a very difficult and challenging task. Earlier work conducted by several researchers revealed significant differences in the measured values of the elastic properties and of the nominal strength of polysilicon without providing in-depth explanations for such a variety of observations. A main question arose, therefore, as to whether the newly evolving test methodologies were adequately precise. In pursuing that question, a round robin study [Sharpe et al., (1998)] demonstrated the inconsistency of measured modulus and strength values, even when specimens from the same source were examined. The material in that work was fabricated in close physical proximity from the same wafer of the same run and in the same deposition reactor at the Microelectronics Center of North Carolina (MCNC, now Cronos-JDS Uniphase). In that round robin effort, the elastic modulus values differed considerably, namely from 132 to 174 GPa, and the strength also demonstrated a rather wide dispersion, ranging from 1.0 GPa, for specimens tested in tension, to 2.7 GPa for specimens tested in bending. A second round robin examination, conducted on material fabricated at the Sandia National Laboratories [La Van et al., (2001)] also showed signs of inconsistent rupture strengths demonstrating a dependence on specimen size and measurement technique. A relatively new round robin test was carried out by [Tsuchiya et al., (2005)]. The specimens, produced with the same process, in the same wafer, were distributed to five different research groups. In this case the obtained modulus varied from 134 GPa to 173 GPa, while fracture strength varied from 1.44 GPa to 2.51 GPa. As a result of these findings, it appears advisable for any micro-fabrication facility not to use the properties cited in the literature for final design and verifications but to identify the most feasible measurement technique and to conduct measurements for every fabrication run. However, while

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individual measurements provide values of some effective modulus and strength for a particular material, such measurements can not be identified unequivocally as mechanical properties of polysilicon. Variation of these properties in micro-machined components is often on the order of 50% or larger, frequently with no particularly strong correlation to the silicon deposition process or crystal structure.

3.2. Testing methodologies

Resorting to some definitions provided by ASTM (American Society for Testing and Materials) standards for testing at macro-scale, among material properties of interest in the context of the present discussion are: the Young’s modulus, defined as the slope of the linear part of the stress-strain curve; the Poisson’s ratio, which measures the lateral expansion or contraction when the material is subjected to uniaxial stress in the linear range; the fracture strength, i.e. the normal stress at the beginning of fracture; the response to cyclic loading in terms of the S-N curve, which is a plot of the applied stress versus number of cycles at rupture. In order to measure material properties one should be able to construct a specimen according to a given design, apply an external input in terms of forces or displacements and measure the specimen response using direct procedures, in the sense that the variable of interest should be (almost) directly measured. All these steps are fully standardised at the macro-scale and are currently applied for testing construction materials like steel and concrete. Unfortunately, these practices cannot be easily applied at the scale of MEMS. In particular one has to resort to fully or partially indirect approaches. E.g., in order to measure the Young’s modulus, cantilever beams in bending are often utilised; deflection is measured and the property of interest is computed on the basis of an analytical or numerical model of the beam. Even during on-chip tension tests some sort of inverse analysis has to be performed since, in general, only capacitance variations are measured directly while deformations are obtained on the basis of a numerical model. Many testing methodologies have been proposed in the scientific literature for the extraction of static

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mechanical properties of polysilicon [e.g. Sharpe, (2002)]. Limiting the attention to silicon MEMS, a first general classification of test procedures can be made between off-chip and on-chip devices. In both cases the micro-device is generally produced by deposition and etching procedures. On-chip test devices [see e.g. Corigliano et al., (2004)] are real MEMS in which actuation and sensing is performed with the same working principles of MEMS. On-chip devices rely on the fact that all the mechanical parts needed to load the specimen and the majority of the ones (electrical or optical) needed for the measure of displacements and strains are built together with the specimen during the micromachining fabrication process. Usually, in these structures two main parts (the actuator and the sensing devices) can be found. In many cases these consist in a large number of capacitors that can be used for the creation of an electric field, this in turn causes a force to act onto the specimen or for the measure of a capacitance which is directly related to the displacement of the specimen itself. The advantage of on-chip testing methods is linked to the ease of fabrication and of use (usually without costly equipments) and to the fact that complicate handling and alignment of the specimen are avoided. The major drawback is that the force developed by on-chip actuators can be insufficient to break specimens in quasi-static conditions and that the maximum displacement is also limited, in the order of the micrometer. On-chip testing of MEMS devices is especially advocated since the thin-film microstructure and state of residual stress is a strong function of micro fabrication process steps. Nevertheless it requires accurate modelling and numerical/analytical analyses of the whole device. An off-chip test [see e.g. Sharpe, (2002)] generally resorts to some sort of external gripping mechanism actuating the force and an external sensor measuring the response of the specimen. All the experimental set-ups that use an external apparatus (load cells, micro-regulation screws, etc..) in order to create a stress state into the specimen are usually included in the category of off-chip testing. In this case a lot of attention has to be paid during the handling of tiny MEMS specimens, during the system-specimen alignment and to the specimen gripping systems. The challenge of picking a specimen only a few micron thick, place it into a

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test machine and perform the test is a formidable task. The main advantage of-off chip methodologies is that the forces and the displacements can be relatively high to break even several micrometers thick specimens in pure tension and that many different configurations can be set up to create an a priori desired multi-axial stress state into the specimen. In principle, material parameters for MEMS, and in primis the Young’s modulus E, can be determined exploiting several test devices. Among others: tension tests, bending of cantilever beams, resonant devices, bulge tests, buckling tests. Clearly, the most direct approach is the tension test, but unfortunately this is not always applicable since it requires the deployment of considerable forces at the micro-scale in order to produce sensible deformation in the specimen. Hence a wealth of alternative solutions have appeared in the literature. In the following two Sections the experimental mechanical characterization of polysilicon as a structural material for MEMS is discussed. It was decided to group the most important experiments in two principal families:

• quasi-static testing, used for the characterization of Young’s modulus, Poisson’s ratio and fracture properties [Section 3.3];

• high-frequency testing, used for the characterization of fatigue properties [Section 3.4].

3.3. Quasi-static testing

The very first tests carried out for the quasi-static mechanical characterization of the material can be traced back in the 80’s [Chen and Leipold, (1980)]. The volume of the specimens used were approximately 1 cm3, huge if compared with typical MEMS dimensions. From the first half of 90’s an increasing interest for this problem arose and the consequence was that a large number of test typology was designed. In the following of this section a selection of the test devices and set ups considered as the most common and interesting is presented. The classification is based upon the actuation mechanism and on the way the system response is read.

19

3.3.1. Off chip tension testThis is the most common and important technique [Bagdahn et al., (2003); Chasiotis and Knauss (2002); Chasiotis and Knauss (2003); Chasiotis, (2006); Chung-Seog Oh et al., (2005); Knauss et al., (2003); Sharpe et al. (1997); Sharpe et al. (1999); Sharpe, (2002); Tai and Muller, (1990); Tsuchiya et al., (2005); Tsuchiya et al., (2002)] for the measurement of mechanical properties for MEMS applications. A MEMS specimen is produced and then placed on a testing system, Figure 3.1. Usually the specimen is gripped to the system with the aid of UV curing adhesives or via electrostatic gripping. This is the way to re-create common macro-scale testing techniques at the micro-scale. The displacement is imposed on the specimen by the use of piezo-transducers, with a resolution in the order of the nanometer. The load cells read the applied load with an accuracy of some µN. Specimen displacements can be measured either with the use of optical systems [Chung-Seog Oh et al., (2005)] or using laser interferometry [Sharpe et al., (1997)] or via digital image correlation [Chasiotis, (2006)]. The results obtained using this technique cover the most important quantities for mechanical design with polysilicon. The measure of Young’s modulus and in some cases of Poisson’s ratio [Sharpe et al. (1997)], together with the rupture strength are the most common for all the research groups that worked with off-chip tension testing. Besides, it is important to underline that with this kind of setup it was possible to study the scale effects due to specimen’s dimension and the stress gradient acting in the specimen. Very recently [Chasiotis, (2006)] it was also possible to determine the fracture toughness KIC of polysilicon.This result was achieved by means of a nano-indentation nearby the specimen that caused a crack to propagate through the substrate and partially involve the specimen. At the end of the process it was thus possible to have a pre-cracked specimen, necessary for a complete fracture mechanics characterization.Figure 3.2 shows other possible configurations for off-chip tension tests [Sharpe, (2002)]. In Figure 3.2(a) a tensile specimen is first patterned onto the surface of a wafer and then the gauge section is exposed by etching of

20

the wafer. The larger ends are gripped in a test machine. In Figure 3.2(b) a specimen is fixed to a die at one end and actuated by means of an electrostatic probe at the other end.

Figure 3.1. Typical scheme of a tension test. (From Tsuchiya et alii, 2002)

Figure 3.2. Off-chip tension test specimens: (a) supported in a frame; (b) off-chip tension test specimen fixed at one end (From Stanley et alii, 2002)

3.3.2. Off chip and on-chip bending testOut of plane bending of test specimens is generally performed via an off-chip apparatus as in Figure 3.3 where a cantilever beam is deflected by a diamond stylus. The deflection of the free end is measured and the Young’s modulus is obtained through inverse modeling of the cantilever beam. However, if the beam is long, forces are small and difficult to calibrate; if the beam is short, forces are higher but inverse analysis of the

21

Figure 3.3. Off-chip out-of-plane bending of a cantilever beam (From Hollman et alii, 1995)

beam is more involved. Doubly supported silicon-micromachined beams can also be used to study the out of plane bending of materials (see Figure 3.4). In this case a voltage is applied between the conductive polysilicon or micromachined beam and the substrate to pull the beam down. The voltage that causes the beam to make contact is a measure of the beam stiffness. Residual stresses in the beams and support compliance cause significant vertical deflections, which affect the performance of these micro-machined devices. Tests need to be supported by models of the devices that takes into account the compliance of the supports and the geometrical nonlinear dependence of the vertical deflections on the stress in the beam.

Figure 3.4. On-chip out-of-plane bending (From Kobrinsky et alii, 1999)

In-plane bending is a classical test for MEMS since several structural parts of accelerometers are subjected to this kind of deformation. A typical on-chip layout is presented in Figure 3.5, where a cantilever polysilicon beam attached to a moving mass (on the left) is subjected to bending induced by the fixed rectangular block (on the right). Actuation is performed by

22

means of interdigited comb-finger capacitors. This test is often conducted to establish the flexural strength of the cantilever beam as a structural component, but requires considerable care when employed for the evaluation of the Young’s modulus due to the uncertainties in geometrical parameters and model of the beam. Additional informations and a wide discussion on the on-chip approach can be found in Chapter 5.

Figure 3.5. On-chip in-plane bending (Courtesy of MEMS Production Divisionof STMicroelectronics)

3.3.3. Test on membranes (Bulge test)The bulge test is one of the earliest techniques used to measure the Young’s modulus, Poisson’s ratio and/or residual stress of non-integrated, free-standing thin structures. This testing method relies on the use of thin polysilicon membranes (circular, square, or rectangular in shape, see e.g. Figure 3.6), relatively easy to design and realize, bonded along their periphery to a supporting frame. [Jayaraman et al., (1998); Tabata et al., (1989); Yang and Paul, (2002); Ziebat, (1999)]. Microfabrication techniques are particularly well suited for the creation of such test structures with reproducible and well-defined boundary conditions. During the test the membrane is loaded with a pressure difference acting on the top and bottom surfaces. The membrane deforms

23

and its profile is measured with a profilometer. Usually the deflection at the centre is recorded as a function of the applied pressure. Several analytical or semi-empirical formulas exist correlating deflection to elastic properties. From this test it is possible to measure:

• the biaxial elasticity modulus;• the Poisson’s ratio;• the nominal rupture strength;• the internal stresses (measuring the buckled configuration of the

membrane);

Figure 3.6. Typical scheme of a bulge test. (From Ziebat, 1999)

One of the shortcomings of this methodology is that sometimes the membranes separates from the substrate before the end of the test. Moreover, since the mechanical response of the membrane varies with the third power of its thickness and the fourth power of its lateral dimension, it is necessary to have a good fabrication technology and an accurate measure of the thickness of the layer.

3.3.4. Nanoindenter-driven testNanoindenters are often used for the mechanical characterization of thin films. Basically there are two ways of using nanoindenters: film nanoindentation and the use of a nanoindenter as an actuator to load MEMS structures. Hardness (indentation) tests are routinely used to characterize large-scale structures. In direct analogy, considerable efforts have been made to develop nanoindentation techniques to characterize microscale structures, and commercial instruments have been developed [e.g. Li and Bhushan, (1999)]. Nanoindentation experiments [Xiadong Li and Bhushan, (1999); Ding et al., (2001); Kim et al., (2002); Chung-Seog Oh

24

et al., (2005)] are performed allowing the tip to penetrate the film under study. During the penetration in the layer, an elastic plastic stress state is generated. This is the main reason why the elastic characterization is done during the unloading phase, when the tip starts going backward to reach the rest position. The results of this experimental test is a force vs. penetration depth plot (Figure 3.1) and the projected area of contact under the indenter. Young’s modulus and fracture properties are computed using some semi-empirical formulas. The main advantage of this method is that there is no need for an ad hoc designed specimen; it is sufficient to have a portion of material large enough to apply the nanoindenter. However, the application of this technique to thin films is complicated by several factors including substrate effects and pile-up of material around the indenter. Another possibility is to use the nanoindenter tip as an actuator to perform a sort of off-chip test (Figure 3.7). In these tests the tip moves an extremity of the specimen, causing a stress state in it. The applied load is measured with a piezo scanner, while there are different ways to measure the displacement of the specimen, like interferometry [Espinosa et al., (2003)] or the measure of the displacement of the tip with aid of a laser beam and photodiodes [Sundarajan and Bhushan, (2002)]. This methodology could be very accurate, but it needs a very expensive instrumentation.

Figure 3.7.Force vs displacement plot of a nanoindentation. (From Xiadong Li and Bharat Bhushan, 1999).

25

Figure 3.8. Typical AFM driven test. (From Sundarajan and Bhushan, 2002).

3.4. High frequency testing

Polysilicon in MEMS technology is used for the fabrication of resonators, gyroscopes and other devices that oscillate at high frequencies during their whole life. One of the most important failure mechanisms for such systems is fatigue. Fatigue is usually interpreted as a phenomenon caused by the motion of the dislocations present in the material that can coalesce during the stress cycles to form micro-cracks. Micro-cracks then join together to form one or more macro-cracks that cause the failure of the structure. Polysilicon is a brittle material and there is no dislocation motion under temperatures of about 900° C, therefore it is not expected to be prone to fatigue in usual operating conditions. Nevertheless, in the second half of ‘90s some groups in the US started working on this subject and found that also polysilicon can undergo fatigue after a large number of cycles, typically more than 109, combined with high stress levels. In

26

order to reproduce experimentally fatigue failures, it is very important to work with experimental set ups that can allow a relatively high frequency testing (at least 1kHz) which in turn allow to reach a large number of cycles in reasonable time. On-chip tests are usually the best choice for this kind of study because they can work at high frequencies and due to the fact that with some electrical control system it is quite easy to perform multiple fatigue tests at the same time.

3.4.1. Fatigue testing and on-chip structuresAs pointed out in the previous sections, on-chip test systems make in general use of electrostatic actuation between a fixed (stator) and a movable part (rotor) to load the specimen with a desired level of stress. The force developed by the actuator is proportional to the actuation area and inversely proportional to the gap between the rotor and the stator. It turns out that to have a force sufficiently large to induce fatigue into the specimens one should have an highly scaled lithography, a thick polysilicon layer and a design area big enough for the thousands of capacitors needed for the actuation. These requirements are the main reason why on-chip testing is not common for quasi-static characterization. Since the MEMS is a dynamic system, the force needed to move the seismic mass decreases if one loads the structure with a time-varying force at a frequency close to the resonant frequency of the system. This is exactly what is usually done in fatigue tests for MEMS, in these cases a reasonable low voltage is used to bring the specimen to fatigue rupture. Very interesting results were obtained e.g. in [Bagdahn and Sharpe, (2003)].

3.4.2. Fatigue mechanismsAs discussed in the previous section, it has been experimentally shown that fatigue in polysilicon is a possible failure mechanism. Nevertheless, the reasons why fatigue rupture occurs are not yet completely clear and understood. Among the most active groups in the study of fatigue in polysilicon, are those of Pennsylvania State University and of Case Western Reserve University. These two groups proposed the most known

27

and accepted interpretations for fatigue mechanisms in polysilicon. The first one after an experimental campaign conducted with specimens like that shown in Figure 3.9, [Muhlstein et al., (2001); Muhlstein et al., (2002); Muhlstein et al., (2004)] proposed a mechanism called reaction layer fatigue (Figure 3.10) which can be summarized as follows: the native oxide is formed, when the polysilicon is first exposed to air, as one of the final steps of the process; the oxide thickens in the highly stressed regions and becomes the site for environmentally assisted cracks which grow in a stable way in the layer; when the critical size is reached, the silicon itself fracture catastrophically by trans-granular cleavage.The second group [Kahn et al., (2000); Kahn et al., (2002); Kahn et al., (2004)] after experiments conducted on specimens like the one shown in Figure 3.11, showed that even an high stress state cannot cause an appreciable growth of the native oxide, thus excluding the pure environmentally assisted fatigue. On the other side, the Authors noticed that increasing the level of humidity, fatigue life decreases. They did not propose a specific fatigue mechanism for polysilicon.Fatigue testing remains one of the most open research area in the field of mechanical characterization of polysilicon. The mentioned researches are only examples of a wider discussion now active in the scientific literature [see e.g. also Ando et al., (2001)].

Figure 3.9. Test structure adopted in (From Muhlstein et alii, 2001)

28

Figure 3.10. Reaction layer fatigue mechanism. (From Muhlstein et alii, 2002)

Figure 3.11. Test structure adopted in (Kahn et alii, 2000)

29

Chapter 4

Weibull theory applied to the study of polysilicon strength

4.1. General concepts

Brittle materials, such as materials used in MEMS applications, together with common engineering ceramics, rocks and concrete have been widely used for structural components for their excellent resistance to heat, corrosion, and wear. But brittle materials also break easily and their strength, i.e., the maximum stress they can withstand, varies unpredictably from component to component even when nominally identical specimens are tested under the same conditions. This variability is not the result of a wrong preparation of specimens, but it is an intrinsic characteristic of the material that cannot deform plastically under the action of growing loads. Therefore, the strength of a brittle material is not a well defined quantity and has to be described with respect to fracture statistics. Since the 70's Weibull distribution [Weibull, (1951)] was widely and successfully used by a number of engineers and scientists for the study and characterization of the strength of brittle materials [Stanley and Inanc (1984), Bažant (1991)]. This is the reason why it has been recently applied also to the study of rupture phenomena in polysilicon MEMS [Chasiotis and Knauss (2003), Corigliano et al. (2004), Corigliano et al. (2005), Sharpe et al. (1997)]. Weibull theory essentially gives a way to estimate the failure probability of a mechanical system, starting from the computation of the probability of failure of its weakest part, the theory is therefore also known as the weakest link approach. By means of the Weibull approach it is possible to take into account the experimental scatter of

30

strength values typical of brittle materials, the statistical size effect and the dependence of the probability of failure on the stress distribution.

4.2. Basic theory

The application of Weibull approach to a uniformly stressed uniaxial specimen gives the following equation for the probability of failure P f:

P f =1−exp− VV r ⟨−u

0 ⟩+

m , (4.1)

where: V is the volume of the bar, u, 0, m are material parameters and the ⟨ x ⟩+ symbol denotes the positive part of x. The parameter u, often called threshold stress, represents the stress below which the probability of failure is null. The parameter 0 is the value of the stress that added to u gives the failure probability of the 63.2% to break a specimen with a volume V r. m, known as Weibull modulus, is inversely proportional to the spread of the rupture strength of tested specimens. In the weakest link framework, the failure probability of the entire structure is defined as the probability that just one element (the weakest) of the structure fails.If a body is loaded with a uniaxial but non homogeneous state of stress, the expression of the probability of failure takes into account the contribution to the total failure probability of every infinitesimal part. Thus, equation (4.1) can be written as:

P f =1−exp− 1V r∫V⟨ x −u

0 ⟩+

m

dv ' . (4.2)

In the case of a multi-axial, non uniform stress state it is usually assumed that cracks form in the planes normal to the principal stresses 1 x ,2 x ,3 x . Thus the probability of failure is given by:

P f =1−exp− 1V r∫V∑i=0

3 ⟨i x−u

0 ⟩+

m

dv ' . (4.3)

31

The above equation is of general applicability and has been obtained by iterating the hypothesis of statistical equivalence of all elementary parts which constitute the volume V. It is also important to notice that the fracture criterion based on principal stresses in a 3D situation is another strong assumption which could be substituted by another one, giving rise to a different stress function in the integral of equation (4.3). The general expression is here applied under the assumption that parameter u=0, this means that all level of stresses have an influence on the probability of failure. Equation (4.3) is then re-written in a more compact way:

P f =1−exp− 1V r∫V x 0

m

dv ' , (4.4)

having defined the equivalent stress x as:

x ≡∑i=0

3

⟨ i x ⟩+m

1m

. (4.5)

The above relations can be used in order to estimate the probability of failure P f of a given structure or solid once the Weibull parameters m and 0 are known and the elastic distribution of stresses has been computed via analytical formulae or numerical solutions, e.g. by means of the FE method. Parameters m and 0 are usually experimentally determined starting from a series of uniaxially tensile tests on cylindrical specimens of volume V; in this simple case equation (4.1) reduces to:

P f =1−exp− VV r 0

m . (4.6)

Weibull parameters can be identified also from a specimen or structure loaded in a multiaxial situation with a non-uniform stress distribution. Let us re-write equation (4.4) in a form similar to (4.6):

P f =1−exp− 1V r∫V x

0 m

dv '≡1−exp[− VV r nom

0 m

] . (4.7)

32

In (4.7) the nominal stress nom and the function have been defined as:

nom≡maxx

[ x ] (4.8)

x ,V , m≡ 1V ∫V x

nom m

dv ' (4.9)

nom represents a nominal stress in the non uniformly stressed specimen or structure, which acts as a scaling parameter for the elastic response. The function depends only on the normalized stress distribution in the linear elastic response and is therefore independent from the load level.

4.3. Statistical size effect and stress gradient effect

One of the main features of the Weibull approach relies on the fact that it is capable to predict many typical aspects of the mechanical behavior of brittle materials. In the MEMS scientific community, many researches were performed in order to understand the importance of the size on the value of the maximum stress sustainable by the structure [Ding et al. (2001)] as well as the importance of the stress distribution inside the mechanical component [Jadaan et al. (2003)], that can have a big impact on the value of the peak load that causes the collapse. In order to compare the behaviour of different structures, it is possible to define a critical stress level 0

* that gives the 63.2% of failure probability for a body under the action of assigned loads. From equation (4.7) it follows:

VV r0

*

0

m

=1 . (4.10)

Therefore 0* can be written as:

0*=0 V r

V 1m , (4.11)

33

that expresses the dependence of 0* on the parameters , V and m.

4.3.1. Effect of the modulusWeibull modulus m can be considered as a measure of the spread of the failure stress around its mean value. As shown in Figure 4.1, this value governs the slope of the failure probability plot in its linear part. If the value of the modulus is small, the failure distribution presents a big spread. As the modulus increases, the probability distribution becomes narrower, until, when m∞, the cumulative probability function becomes a step function. It means that a stress level lower than 0 cannot cause a failure into the structure, while a stress level bigger that 0 for sure causes the mechanical collapse. Weibull modulus affects the importance of the size effect, too. If one takes equation (4.11), once fixed the value of V, V r and , and plots the value of the parameter 0

* against the value of the modulus, as done in Figure 4.2, he notes that for large values of m, 0

* approaches to 0. This means that for large values of the modulus the size effect vanishes.

Figure 4.1. Failure probability with different values of the modulus

34

Figure 4.2. 0*/0 ratio as a function of the Weibull modulus

4.3.2. Effect of the volumeWeibull approach relies on the probability of finding a critical defect in the structure. Therefore it appears natural that a structure with a larger volume has a worse mechanical behaviour than a smaller one and mathematically this can be seen considering equation (4.11), plotted in Figure 4.3.

Figure 4.3. 0*/0 ratio as a function of the volume

35

Figure 4.4: Schematics of a pure tension on specimens with different volume

To make an example it is possible to consider two structures, as schematized in Figure 4.4, with respectively a volume equal to V and 3V, loaded with the same uniaxial tensile stress . Since the stress is constant in every point of the volume, the value of the parameter , computed applying equation (4.9), is one. Considering then the same value of the representative statistical volume V r for both structures, it is possible, making use of equation (4.6), to compute the failure probability in the two cases, as shown in Figure 4.5. The plots indicates that, fixing the level of the applied stress, the size effect associated to the volume is interpreted as a bigger failure probability of the larger structure with respect of the smaller one.

Figure 4.5.Failure probability of the structures of Figure 4.4

4.3.3. Effect of the stress distributionThe stress distribution in a brittle structure plays a very important role in determining the maximum load applicable. Let us for instance consider

36

two different load conditions schematically drawn in Figure 4.6. The left one represents a specimen with an homogeneous state of stress (pure tension) and the right one represents a specimen with a non uniform stress field (bending cantilever) with the same maximum value of the former case.

Figure 4.6: Schematics of a pure tension and a pure bending specimens

In a pure tensile loading condition, every single material point contributes to the total failure probability, while in the bent cantilever the contribution is given only by a small portion of the body, near to the clamped edge, that is highly stressed. It turns out that the failure probability of the single tension specimen is higher than that of the bending specimen. In the Weibull approach, the parameter that takes into account the stress distribution is . In the two scenarios considered, using the definition given in equation (4.9) and fixing the values of the modulus and of the volume, the value of in the pure tension case is:

tension=1 . (4.12)

In the case of the bending cantilever, using the Euler-Bernoulli beam model, disregarding the shear stresses and considering the reference frame of Figure 4.7, the positive tensile stress in a generic point P≡x , y of the body is:

x , y =F L−x I

⟨ y ⟩+ , (4.13)

where F is the applied force, L the is length of the beam and I is the moment of inertia of the section.

37

Figure 4.7: Reference frame for the cantilever beam

The maximum value of the stress is therefore:

nom=F L

Ih2

, (4.14)

where h is the height of the section.Applying the definition of given in equation (4.9), it yields:

=1V ∫V x , y

nom m

dv '= 1L t h∫0

L

∫0

h2 [t L−x

L2yh

m]dx dy , (4.15)

having defined t as the thickness of the beam. The final value of , computed solving equation (4.15) is:

= 1m122m1 . (4.16)

Equation (4.12) can be used to compare the failure probability of the pure tension structure against that of the bending cantilever. The value of the stress that causes a failure probability equal to 63.2%, upon the substitution of the value of in equation (4.7) in the pure tension case is:

0*tension=0 V

V r − 1

m , (4.17)

while for the bent beam is:

0*bending=0 V

V r − 1

m [m12 2m1]1m . (4.18)

38

Combining equations (4.17) and (4.18), one can express the ratio between the critical stress 0

*bending and 0*tension :

0*bending

0*tension =[m12 2m1]

1m . (4.19)

This function, plotted in Figure 4.8, shows that the stress necessary to have a fixed value of failure probability for the bent beam is always bigger than the one for the purely tensioned structure. Figure 4.9 is a plot of three failure probability curves. The continuous line refers to the pure tension test and the dashed curves refer to a bending cantilever with two values of Weibull modulus.

Figure 4.8: Values of 0

*bending

0*tension as a function of the Weibull modulus

39

Figure 4.9: Influence of on the failure cumulative plot

4.4. Application of Weibull approach to 3D structures

The computation of the failure probability of a structure or the determination of Weibull parameters on a pure tension specimen or on a simple cantilever beam are relatively easy, a closed form solution in fact exists for expression of the stress field. When the shape of the structure and the loading conditions do not allow for any simplifying hypotheses in order to achieve a closed form solution, these operations may present some practical difficulties. For this reason two different routines were implemented. The former one makes use of the results of a linear elastic FE solution to compute numerically the effect of the volume and that of the stress distribution in the structure. The latter needs both the FE solution to compute the value of the parameter and the experimental results for the minimization of the objective function that allows for the determination of Weibull parameters.

4.4.1. Failure probability computationMost of the effort necessary for the computation of the failure probability is due to the computation of the coefficient (Equation ), that is a

40

measure of how in-homogeneous is the stress inside the body. Once discretized the body into Finite elements, from equation (), making use of the linearity property for the integral operator, the equation that gives the value of coefficient can be rewritten as:

≡1V ∫V x

nom m

dv'= 1V ∑i=0

n.el

∫V el

iel xnom

m

dv ' , (4.20)

where n.el is the total number of elements of the discretization, iel x is

the stress norm into the i-th element and V el is the volume of the i-th element. Using Finite element solution in terms of stresses, equation (4.20) can be approximated as:

≈ 1V ∑i=0

n.el

∫V el

iFE x nom

m

dv ' , (4.21)

where iFE x is the approximated stress field into the i-th element.

Then, solving numerically the integral in (4.21) by the means of the Gaussian integration, the final expression of becomes:

≈ 1V ∑i=0

n.el

∣J i∣∑j=1

n.gp

w j elFE x j nom

m

, (4.22)

Where ∣J i∣ is the determinant of the Jacobian of the i-th element, n.gp is the number of Gauss points and w j is the j-th Gauss weight.Since the Weibull modulus m can be even in the order of fifty, the number of Gauss points needs to be as large as possible. As an example in this work the number of Gauss points for the integration of quadratic tethraedra was chosen to be 11.Once computed , the expression of the failure probability as a function of the maximum value of the stress norm nom is given by equation (4.7).

4.4.2. Weibull parameters identificationThe identification procedure is composed of two separate steps. The first one consists in the identification of the modulus m and of the structural parameter 0

*. In order to accomplish this, a least square minimization is

41

performed on an objective function. Being the experimental data the couples of values P f i

exp , iexp, the objective function to minimize is:

m ,0* = ∑

i=1

numtest

[P f iexp−P f i

exp ]2 . (4.23)

After the minimization of , once obtained m and 0*, it is possible to

compute the value of the parameter as shown in the previous paragraph. Then, manipulating the equation (4.11), the expression of the material parameter 0 becomes:

0=0* V

V r1m (4.24)

Since the right hand side of the (4.24) is known, it is finally possible to compute 0.

42

Chapter 5

On-chip testing of ThELMATM

polysilicons

5.1. The ThELMATM process

The test devices have been produced following the surface micromachining process ThELMA (Thick Epipoly Layer for Microactuators and Accelerometers) which has been developed by STMicroelectronics to realize in-silicon inertial sensors and actuators.The Thelma process permits the realization of suspended structures anchored to the substrate through very compliant parts (springs) and thus capable of moving in a direction orthogonal to the plane of the wafer, such as the structure described in paragraph 5.2 or in a plane parallel to the underlying silicon substrate, such as the one described in paragraph 5.3. The process flow exploits several state-of-the-art integrated circuit technology steps, together with dedicated MEMS operations, like high aspect ratio (trench), dry etch and sacrificial layer removal for structure release.This technology is more complex than the Surface Micromachining but allows to obtain silicon structures with a relatively large thickness; this, in turn, increases the vertical surfaces and the global capacitance in electrostatic actuators which move parallel to the substrate.The process consists of the phases concisely enumerated hereafter and schematically illustrated in Figure 5.1.

43

Figure 5.1. Schematic illustration of Thelma surface micromachining process. (1) Substrate thermal oxidation. (2) Deposition and patterning of horizontal

interconnections. (3) Deposition and patterning of a sacrificial layer. (4) Epitaxial growth of the structural layer (thick polysilicon). (5) Structural layer patterning by

trench etch. (6) Sacrificial oxide removal and contact metalization deposition.

44

• Substrate thermal oxidation. The silicon substrate is covered with a 2.5 µm-thick layer of permanent oxide obtained with a thermal treatment at the temperature of 1100 °C.

• Deposition and patterning of horizontal interconnections. The first 0.7 µm-thick polysilicon layer (poly1) is deposited above the thermal oxide; this layer is used to define the buried runners which are used to bring potential and capacitance signals outside the device, or eventually as a thin structural layer.

• Deposition and patterning of a sacrificial layer. A 1.6 thick oxide layer is deposited by means of a Plasma Enhanced Chemical Vapor Deposition (PECVD) process. This layer, together with the thermal oxide layer, forms a 4.1 µm-thick layer which separates the moving part from the substrate and which can be considered analogous with the sacrificial layer in a Surface Micromachining process.

• Epitaxial growth of the structural layer (thick polysilicon). The polysilicon is grown in the reactors, thus reaching a thickness of 15 µm.

• Structural layer patterning by trench etch. The parts of the mobile structure are obtained by deep trench etch which reaches the oxide layer.

• Sacrificial oxide removal and contact metalization deposition. The sacrificial oxide layer is removed with a chemical reaction; in order to avoid stiction due to attractive capillary reactions, this is done in rigorously dry conditions. The contact metalization is deposited; this will be used to make the wire-bonding between the device and the metallic frame.

45

Figure 5.2. Section view of epitaxial polysilicon

The final structure of silicon produced by the Thelma process is given by a series of single crystals in the shape of vertical columns (see Figure 5.2), each of them grown in an epitaxial reactor.

5.2. Study of the role of grain structure and surface defects in thin polysilicon

5.2.1. Out of plane structure descriptionThe first on-chip device discussed in the present work is shown in Figure 5.3(a) and (b) [see also Cacchione et al., (2005)]. Figure 5.3(c) is a zoom of the central part where the 0.7 µm thick beam specimens are placed and shows the deformed shape of the specimens caused by the application of the force. The actuator is an holed plate of 15 µm thick polysilicon and it is suspended on the substrate by means of four elastic springs placed at the four corners (see Figure 5.4(b)).

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Figure 5.3. (a) Sem image of the structure; (b) top view of the structure; (c) specimens working principle

Figure 5.4. (a) Cross section next to the specimens; (b) sensor and suspension spring detail

The holed plate is also connected to the thin polysilicon film specimens placed at the centre, as shown in Figure 5.4(a). The two symmetric specimens are in turn connected on one side to the holed plate, while on the other are rigidly connected to the substrate, therefore behaving as a couple of doubly clamped beams. The holes in the plate are due to the

47

etching process for the elimination of the sacrificial layer, thus allowing for the movement of the plate with respect to the substrate. The movement in the direction orthogonal to the substrate is obtained by electrostatic attraction. The application of a voltage on the plate produces an electrostatic force that pushes the plate toward the substrate, so that the whole plate and the substrate work as a big parallel plate electrostatic actuator. When the plate moves towards the substrate, the couple of specimens undergoes bending, as shown in Figure 5.3(c). It is important to remark that only the squared part of the holed plate acts as an actuator, while the holed rectangular parts added to each side of the plate act as sensors (see Figure 5.4(b)). These sensors are designed in order to measure the capacitance variation with respect to rectangular electrodes placed under the thick polysilicon mass. From the capacitance variation is then possible to compute the displacement of the actuator in the direction orthogonal to the substrate. The length of each specimen is 7 µm; in order to force the rupture in a priori chosen section, as shown in Figure 5.3(c), their cross section changes with a linearly varying width which decreases from 3 µm to 1 µm.

5.2.2. Numerical models for the computation of electrostatic fields

Given the complex geometry of the actuator and of the sensing system, it was not possible to use simple analytical formulae to compute the force developed by the plate and the displacement measured by the sensing system in order to achieve the final force vs. displacement experimental plot. For this reason a series of numerical analysis was performed.

ActuatorFE simulations were done to understand the electrostatic behavior of the plate, the influence of the holes and of the fringing perimetrical fields on the force developed by the actuator.The analyses were carried out varying the gap between the plate and the substrate, in order to get the force vs. gap relationship. The direct result of the analysis was the electrostatic energy of the system. The numerical

48

results were then fitted by means of a 6th order polynomial. The force developed by the actuator and the energy contained in the electric field are linked by the equation:

Fact=−∂ E∂ x , (5.1)

where x is the displacement variable and E is the electrostatic energy. Therefore computing the derivative of the polynomial expression of energy with respect to the displacement, it is possible to compute the force.These results were compared with the analytical formula for the electrostatic force developed by an infinitely thin square plate, without holes, with the same lateral dimensions, taking into account perimetrical fringing effects. As it is possible to see in Figure 5.5, in the gap range available for the actuation (4.2÷1.8 µm), the two curves are almost superposed, thus allowing to use a very simple analytical formula to compute the force Fact developed by the plate:

Fact=120[ l 2

g0−x 2 P

21

g 0−x ]V 2 , (5.2)

where 0 is the permittivity of vacuum, l the side of the plate, V the applied voltage, P the perimeter of the plate, g0 the gap at rest and x the displacement. The results of the simulations show that the holes present in the plate reduce the amount of the net surface of the capacitor but, due to the non-negligible lateral dimensions of the sidewalls, the electrostatic field inside the holes is not null and consequently compensates the lack of net surface of the holed plate.

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Figure 5.5. Actuation force vs. gap plots

SensingThe distance between the rotor and the stator part of the sensing electrodes is 1.6 µm when zero voltage is imposed between rotor and stator; it decreases untill 0.2 µm when the maximum voltage is applied.

The surface of the sensing electrodes is 36 x 157 µm2. This means that if one would like to use FE to simulate the system when voltage is high, he would be forced to discretize the system with a very fine mesh in order to avoid high aspect ratios in the elements.

Figure 5.6. Bem mesh of the capacitive sensor

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This is the main reason why the Boundary Elements Method implemented in the commercial software CoventorwareTM was used. This method needs only the discretization of the surfaces of the electrodes (Figure 5.6), to compute the capacitance of the sensing system and its variation with the gap. The analysis were carried out decreasing the gap with a step equal to 0.1 µm; the results were subsequently fitted (Figure 5.7) to get the expression below:

x= 0.2598C 21.3112C−0.0315

, (5.3)

where x, in µm, is the gap and C, in pF, the measured capacitance.

Figure 5.7. Gap versus capacitance variation plot

5.2.3. Data reduction procedureTests were carried out at room temperature and at atmospheric humidity, with a probe station mounted on an optical microscope. The experimental setup is shown in Figure 5.8.

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Figure 5.8. Scheme of the experimental setup

The input voltage given to the structure and the variation of the capacitance induced by the displacement of the rotor is measured connecting an Agilent Precision LCR Meter between two pads. The LCR resolution, in the range of measures of interest in this work, is ±1 fF. A slowly increasing voltage is applied in order to induce quasi-static loading conditions in the specimen. The experimentally determined capacitance vs. voltage plots are transformed in force vs. displacement plots shown in Figure 5.9, by making use of the relationships between capacitance and displacement and between voltage and electrostatic force described in the previous section.

Figure 5.9. Displacement vs. Force plot

Starting from the force-displacement plot, the force acting on the specimens was obtained by subtracting the part equilibrated by the elastic

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suspension springs in the four corners of the holed plate (see Figures 5.3 and 5.4). Then from the net force vs. displacement curve, it was possible to obtain the Young’s modulus of the material and the nominal rupture strength with the aid of a linear elastic Finite Element (FE) analysis performed on the 3D FE mesh of Figure 5.10, as described later on.One half of the specimen is modelled and discretized and a fixed uz displacement is imposed on the anchor point of the specimen to the plate. On the anchor points between the substrate and the oxide underneath the specimen a null displacement is imposed .

Figure 5.10. Finite Element mesh and mesh detail used for mechanical simulations

Making the hypothesis of linear elastic material and small displacements, the global mechanical behavior results to be linear and the specimen’s stiffness is proportional to the Young’s modulus of the thin layer:

F=K sp uz , K sp=E k sp , (5.4, 5.5)

where F is the applied force, K sp is the elastic stiffness and k sp is the stiffness divided by the Young's modulus E. By making the additional hypothesis that relation (5.5) holds also for the real structure, with the same coefficient k sp, it is possible to find the value of Young’s modulus from a linear fit of the experimental force vs. displacement plot, which allows to compute the experimental stiffness K sp. Due to the linear

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proportionality stated by (5.5) it is in fact possible to write:

k sp=K sp

FE

E FE =K sp

Ex

E Ex , E Ex =E FE K spEx

K spFE . (5.6, 5.7)

where the superscript (FE) refers to the simulation and the superscript (Ex) to the measured quantities. The determination of the rupture strength was based on the same hypotheses made to find relations (5.4 and 5.5). The same linear proportionality between the applied force and the maximum stress in the specimen can be stated both for experimental and for simulated results:

F zFE

maxFE=

F zEx

maxEx , max

Ex =maxFE F z

Ex

F zFE . (5.8, 5.9)

Relation (5.9) was applied in order to find the maximum stress at rupture, substituting to F z

Ex the experimental value of the force at rupture, to F zFE

the value of the force equivalent to the applied load in the FE analysis and to max

FE the maximum value of tensile stress found in the FE simulation, in the section where rupture occurred experimentally (Figure 5.11). It is important to remark that, besides the hypotheses of homogeneity and linearity, relations (5.6, 5.7) and (5.8, 5.9) hold only if the geometry of the specimen is carefully reproduced in the FE model. The final FE mesh was therefore obtained after SEM images in order to reproduce the real geometry.

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Figure 5.11. Principal tensile stress contour plot

5.2.4. Results and discussionThe values of Young’s modulus obtained from 20 tests are 174 ± 9 GPa, with a variation coefficient of 5%. The low dispersion around the mean value confirms the quality and good reliability of the whole procedure. The data concerning rupture of the specimens were interpreted in the framework of Weibull approach, as discussed in Chapter 4. The Weibull modulus obtained was m=5.13 and the Weibull stress 0

* obtained directly on the non uniformly loaded specimen is 2773 MPa with a standard deviation equal to 648 MPa; the Weibull stress 0 obtained after the reduction due to the stress gradient and size effect, i.e. the Weibull stress of an equivalent, uniformly loaded, reference volume specimen is 2237 MPa. These results have been compared to the ones obtained with in plane moving structures made up with the same LPCVD 0.7 µm thick polysilicon [Cacchione et al., (2004)]. In this case the values obtained for the Young’s modulus were 178 ± 2 GPa and the Weibull parameters resulting from the data reduction were m=6.18 and 0=1840 MPa. The value of the Young’s modulus remains substantially the same in the two cases, meaning that the elastic properties of the material can be considered constant along the thickness of the deposited layer and that in this case there is no important influence of the crystalline granular

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structure. Something different happens analysing the rupture properties of the material. In fact the Weibull modulus remains almost the same, while the value of 0 is about 20% bigger in the case of out of plane loading condition. Figure 5.12 shows that trying to predict the failure probability of the “out of plane structure” using the data obtained from the in plane one, the result would be a little conservative.

Figure 5.12. Prediction of the failure probability of out of plane structures using in plane failure data

The causes of this difference could be related to the fabrication process and in a special way to the thin polysilicon etching step that defines the geometry of the specimen itself. In fact the polysilicon sidewalls surfaces are exposed to the gas during the etching, while the top surface does not, being it covered with the resist mask. The etching could introduce on the sidewalls surface micro flaws, micro defects and internal stresses, only partially recovered during the subsequent process steps, which could reduce the ultimate strength of the material.

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Figure 5.13. A Sem image of the sidewalls

Figure 5.13 is a SEM image of the sidewalls of the polysilicon layer. It is possible to see that the surfaces are characterized by the presence of micro-grooves. These could be privileged sites for the fracture initiation because of the stress concentration effect they can induce given their relatively small fillet radius.

5.3. Design and testing of a new test structure for the mechanical characterization of thick polysilicon

5.3.1. Structure descriptionThe second device designed and then experimentally tested during this work is shown in Figure 5.14. Compared to the one presented in the previous paragraph, this structure presents many differences. It was designed to test the mechanical properties of the epitaxial polysilicon, the 'thick' layer deposited with ThELMATM process. This film is almost twenty times thicker than the poly1 one and therefore it is necessary a bigger force to break the specimen. This explains the reason for the relatively big dimensions of the structure, that takes up a 1600 by 2250

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Figure 5.14. Top view of the structure and zoom of the specimen.

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µm2 rectangular area. The actuation movement is in a plane parallel to the one of the wafer and it is due to a number of comb finger capacitors, a very common choice for MEMS structures. In the next sub-sections the structure will be divided into four parts, separately discussed in detail.

ActuatorThe electrostatic actuation is realized by more than four thousands comb finger actuators. The comb fingers are grouped on specific structures called arms (see Figure 5.15).

Figure 5.15. Actuation arm (the dashed area represents the fixed part).

Every arm contains 31 comb finger actuators and its capacitance, as a function of the seismic mass displacement x is:

C arm=C031C comb=C 03120 t

gx , (5.10)

where C 0 is the capacitance at rest of the system, C comb is the capacitance of a single comb finger actuator. The symbol t represents the thickness of the layer, equal to 15 µm, and g, equal to 2.2 µm , the gap between stator finger and rotor finger. The force developed by every arm is:

Farm=12∂Carm

∂ xV 2=31

0 tg

V 2 , (5.11)

where V represents the applied voltage. The total number of arms na is 130. Hence the total force developed by the actuator can be expressed as:

Fact=na12∂C arm

∂ xV 2=4030

0 tg

V 2 . (5.12)

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FrameThe frame is a suspended structure that supports the actuator arms (see Figure 5.16). In the central part of the frame six suspension springs (colored in light gray in Figure 5.16) are placed. These springs avoid the collapse of the structure onto the substrate during the actuation. They are rectangular cross-sectioned l=291 µm slender beams. The in-plane width is w=3.2 µm, while the out-of plane thickness is t=15 µm. By assuming a Young's modulus E=145 GPa, as determined in Corigliano et al. (2004), the linear stiffness of the six springs for a movement parallel to the substrate as shown in Figure 5.16 can be easily computed:

k spring=612 EJl 3 =612 E t w3

12 l3 =17.35 Nm

, (5.13)

being J the moment of inertia of the section. With reference to Figure 5.14, the upper part of the frame is clamped to the specimen, that is thus loaded with the force developed by the actuator that is not absorbed by the spring system. It is worth noting that the suspension system was designed in order to be as compliant as possible to leave almost all the force ( the 90% of the force developed) to load the specimen.

Figure 5.16. Schematic of the frame (filled with dashed line). In black the anchor points, in gray the six suspension springs

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SensingIn the upper part of the structure, as in Figure 5.14, there is the sensing system. It is made up with six arms with 80 comb finger capacitors on each (Figure 5.17).

Figure 5.17. Sensing arm (the dashed area represents the fixed part).

The total sensing capacitance, as a function of the displacement of the specimen is:

C sens=C0 sens48020 t

gx . (5.14)

With the aid of FE electrostatic analysis a simulation of the sensing system was performed. The results of the simulations show how the analytical formula holds for a displacement up to 10 µm, that was never reached during the experimental campaign.

SpecimenThe specimen of the structure was designed in order to perform both quasi-static and fatigue testing. It consists in a lever system that causes a stress concentration in a very localized region (Figure 5.18). The specimen can be divided into four parts :

• a beam that is the physical link between the frame and the specimen;

• the lever, that transforms the axial action coming from the beam into a bending moment acting in the notched zone;

• a notch, that is the most stressed part, where the crack nucleates;• a part fixed to the substrate.

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Figure 5.18. Deformed shape of the specimen and contour plot of principal tensile stress of the notched zone

5.3.2. Data reduction procedureThe experimental setup and the instrumentation used is the same as that of the out of plane thin polysilicon structure. In this case the electrical scheme is slightly different because the actuator is used only to load the specimen, while the displacement is measured with the sensing system (see Figure 5.19). This configuration was chosen in order to avoid that any possible deformation of the frame during the test could affect the measure and to measure the displacement as close as possible to the specimen’s area. The measured capacitance variation is used to compute the displacement imposed on the extremity of the load beam by equation (5.10).

Figure 5.19. Electrical scheme of the experimental setup

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This causes the rotation of the lever arm and the creation of the desired state of stress in the notch. The force produced by the actuator for every imposed voltage is computed using equation (5.12). From this information it is then possible to plot the force versus displacement curves (Figure 5.21). Then, following the same procedure described for the out-of-plane structure, it was possible to combine experimental tests and Finite Element simulations to obtain the values of the Young's modulus measured and the values of the maximum tensile stress occurred in every test.

5.3.3. Result and discussionA number of 31 structures, deposited on the same wafer, were tested. As it is possible to see in Figure 5.20, the measures were very repeatable. The force versus displacement plots shown in Figure 5.21 appear to be linear, implying that the electrostatic behavior of the rotor and sensor parts are correctly described by the analytical formulae used in the data reduction procedure. From the slope of the force versus displacement plots it was possible, as done with the out of plane structure, to compute Young's modulus values. The values reduced are in agreement with the ones obtained in [Corigliano et al., (2004)], confirming the overall quality of the data reduction procedure.

Figure 5.20. Capacitance variation versus applied voltage plot

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Figure 5.21. Force versus displacement plots

The mean value measured is 143 GPa, with a standard deviation of ±3 GPa. Even in this case, the data concerning rupture of the specimens were interpreted in the framework of Weibull statistics, as discussed in Chapter 4. As it is possible to notice in Figure 5.22, the experimental results are clearly well interpolated by the Weibull cumulative distribution function. The parameter identification allowed for the computation of the Weibull modulus, that is m=25.76 and the Weibull stress 0. This value, representing the level of stress that gives the 63.2% of failure probability for a pure tension specimen with the same size as the reference volume, is 3622 MPa.

Figure 5.22. Weibull plot of the experimental data (asterisks) and interpolating Weibull cumulative distribution (dashed line)

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After the testing, the specimens were investigated using an optical microscope. The images show that the fracture starts from the notch as predicted from the FE simulations carried during the design phase. From the pictures shown in Figure 5.23, two main aspects can be caught:

• the crack path is often irregular and can be quite different from one structure to another. This can be due to the crystalline structure and grains orientation in the notched area. The grain morphology and orientation is different from one structure to another and has a very important impact on the crack propagation direction;

• the crack starting point is not always the same. This is due to the non uniform flaw distribution on the notch surface, caused by the fabrication process. Flaw distribution is supposed to be responsible of the scatter of the experimental fracture results.

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Figure 5.23. Optical microscope images of broken specimens.

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5.3.4. Estimation of fracture parameters

Approximate evaluation of the energy release rateFrom Finite Element elastic simulation it was possible to approximately compute the critical value of stress intensity factor.

Figure 5.24. Finite Element mesh of the notch and J integral paths.

Considering the notch as it was a real crack, by the means of an internal routine of the Ansys Inc. Finite Element code, a path for the J-integral computation was defined (see Figure 5.24). The code then computes numerically the value of J a for an assigned value of the external force Fa=180N. Three different paths were used to compute J a, as shown in Figure 5.24, with a negligible difference of the three computed values, that resulted 1.282N / m. The value of the J-integral is proportional to the square of the value of the applied force. If one assumes that the proportionality coefficient is the same both for the Finite Element analyzed test structure and for the experimentally tested ones (similarly to what done for the Young's modulus and rupture stress determination), he can obtain a formula that allows for the computation of J for every assigned load F. That is:

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J=J a FF a

2

. (5.15)

The maximum value of the applied force Fc corresponds to the force applied when crack propagates throughout the body. In Linear Fracture Mechanics the J-integral coincides with the energy release rate G c, therefore in a fracture test, the value of J computed at the crack initiation coincides with G. In our case it would not be the exact value because the notch is not a real mathematical discontinuity, as a crack should be, but it can be still used to obtain an approximated value. The value thus obtained from the experimental tests of the approximated energy release rate is:

G c=38.70±3.60 Nm

. (5.16)

Using the plane strain hypothesis, the stress intensity factor can be computed by means of the relation:

K IC= E1−2 Gc . (5.17)

Substituting the values of equation (5.16) and assuming E=143 GPa and =0.2, from (5.17) one obtains:

K IC=2.41±0.11 MPam . (5.18)

These results are reasonable in comparison to the ones obtained by Tsuchiya [see T. Tsuchiya et al., 1997] who reported a value of K IC in the range 1.9÷4.5 MPa m and Ballarini [Ballarini et al., 1997; Ballarini et al., 1998], who measured the energy release rate in polysilicon specimens with values from 19 to 63 N /m.

Fracture propagation Finite element simulationsThe value of the critical energy release rate obtained above from the approximate evaluation was used to simulate the fracture propagation process with the numerical procedure discussed in Chapter 7. The

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simulation was carried out imposing the displacement to the load beam, in order to capture a possible softening behavior of the system. The material model used was linear homogeneous isotropic elastic. The value of the Young's modulus and Poisson's ratio used were respectively 143 GPa and 0.2. The fracture parameters used in the cohesive elements were (see Chapter 7 for details) c=1, =2800 MPa, G c=40 N /m. The imposed displacement ranged from 0 to 7.5 m with steps of 0.03 m. The typical dimension of the process zone near the crack tip for a material with these characteristics is [from Camacho et al., 1996]:

Rp=8

E1−2

G c

2 =0.4 m . (5.19)

In order to have a precise solution of the displacement and stress field near the crack tip it was chosen to have finite elements with a size of 0.1 m, much smaller than the dimension of the zone were cohesive forces act. Several simulations were performed using different meshes in order to be sure to avoid mesh-dependent solutions. The results discussed in the present paragraph refer to the mesh displayed in figure 5.25.

Figure 5.25. FE mesh of the structure and detail in the notched zone.

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Figure 5.26. Force displacement plot from the fracture propagation simulation

Figure 5.27. Crack patterns of the corresponding three parts indicated in the force vs. displacement plot of Figure 5.26

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The main result obtained was the force versus displacement plot showed in Figure 5.26. From the plot it is possible to note that the the mechanical behavior of the structure after the peak load is very brittle. This behavior was indeed encountered in the experimental campaign, where, once reached the peak load, there was no possibility to control the crack propagation and a very fast rupture of the structure was registered from one value of the applied force to the other.From an experimental point of view, such a brittle structural behavior is a limitation if one wants to follow the curve after the peak load. On-chip test structures are in fact used controlling the voltage applied. The voltage, as previously mentioned, creates an electric field and thus a force. But the control of the fracture propagation in a structure that presents softening can be achieved by controlling the imposed displacement.

Charge control of on-chip test structuresOne possible way to have a control on the crack propagation during the softening phase is to inject a fixed quantity of electrical charges into the structure instead of the application of a voltage between rotors and stators. The electrostatic energy stored into the capacitive system, as a function of the capacitance C and of the injected charge Q, is:

E=12

Q 2

C . (5.20)

In a charge controlled system, the electrostatic force Fel acting in the direction x is the opposite of the derivative of the energy with respect to the free coordinate x, i.e.:

Fel=−∂ E∂ x ∣Q=−1

2∂∂ x 1

C Q2 . (5.21)

The partial derivative of the capacitance with respect to the coordinate is:

∂∂ x 1

C = ∂∂C 1

C ∂C∂ x

=− 1C2

∂C∂ x

. (5.22)

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Substituting equation (5.22) into (5.21) it yields:

Fel=1

2C2∂C∂ x

Q2 . (5.23)

The total capacitance of the system can be expressed as the capacitance of every single comb finger (see equation (5.10)) times the total number of comb fingers na present in the structure. Therefore the expression of the force developed by the actuator, as a function of the applied charge Q and of the structure displacement x, obtained by substituting equation (5.10) into equation (5.23), is:

Fel=na0 t g

C 0 g2 na 0 t x 2Q 2 . (5.24)

The electrostatic force developed is counterbalanced by the elastic force Fm due to the MEMS reaction under imposed loads. The equilibrium position of the device xeq can be computed solving the system:

{F el=na0 t g

C0 g2 na 0 t xeq 2 Q2

F m=k x eq xeq

F m=F el

. (5.25)

If the mechanical behavior of the structure is linear, as for instance that of the test structure designed, an increase of the applied charge leads to a new equilibrium position, as shown in Figure 5.28.As it is displayed in Figure 5.29, even if the structure presents a softening branch after the peak value of the force, it is still possible to continue the experimental test without loosing the control of the crack propagation.It is worth noting that the possibility to control the softening branch depends on the value of the capacitance at rest of the system C 0.

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Figure 5.28. Force vs displacement plots for the electrostatic force with different values of charge (in gray); elastic force vs. displacement plot (in black); the circles represent

equilibrium positions

Figure 5.29. Force vs displacement plots for the electrostatic force with different values of charge (in gray); elastic force vs. displacement plot (in black); the markers represent

equilibrium positions

The higher C 0 is, the flattest the electrostatic characteristic, with an assigned value of the charge, will be (see Figure 5.30). Hence the experimental measure of a very fragile behavior of a structure with a

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controllable crack propagation can be achieved only if the value of C 0 is relatively small. The capacitance at rest depends on two main factors:

• the length of the lines that bring the signal from the pads to the structure itself, that is a technological parameter mainly linked to design rules and that can be changed with difficulty;

• the number of capacitors in the actuator. A big number of capacitors, that causes an high value of C 0, is necessary to develop a force sufficient to provoke the crack nucleation and propagation.

Figure 5.30. Force vs displacement plot obtained from fracture propagation simulation and softening branches controllable with different values of C0

In Figure 5.30 is shown a plot of the simulated mechanical response of the structure and a family of three different electrostatic characteristics, that differs for the value of C 0. This parameter was chosen in the range from 0.5 pF to 2.5 pF, being the most probable value in the order of 1.5 pF. It is clear that even if the structure would present a capacitance at rest much smaller than the real one, this method could not allow for a crack propagation controlled experimental procedure.

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Fatigue induced pre-crackA very common technique used at the macro-scale for the execution of a fracture test relies on the use a pre-cracked specimen. Pre-cracking is often obtained by cycling the specimen with a periodic load. This load causes the material to undergo fatigue, and crack originates because fatigue can induce a nucleation and propagation for stresses considerably lower than the fracture stress.As demostrated by various authors [Muhlstein et al., (2001); Muhlstein et al., (2002); Muhlstein et al., (2004); Kahn et al., (2000); Kahn et al., (2002); Kahn et al., (2004)], polysilicon is prone to fatigue (see also Section 3.4). It is then possible to load the structure with a sinusoidal load in order to activate the fatigue mechanism in the notched area and thus create a pre-crack. One possible technique which can be adopted for the control of the cumulated damage into the material is for instance the run-time measure of the structural stiffness. In fact crack propagation into the material causes a very significant stiffness decrease. Figure 5.30 shows, by the means of FE fracture simulations, the stiffness value of the structure vs. crack length.

Figure 5.31. Sfiffness variation as a function of the crack length

Once obtained the pre-cracked specimen, there are mainly two different way to determine the energy release rate.

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The first one can be divided into three steps: • measurement of the crack length; • numerical simulation with linear fracture mechanics models for

the determination of the stress intensity factor as a function of the applied load with reference to the real geometry of the cracked specimen;

• application of a monotonically increasing load and determination of the value for which the crack starts again propagating.

Combining the results of the numerical simulation with the experimental ones it is possible to compute the value of the stress intensity factor and therefore of the energy release rate.The second method that could be used relies on the knowledge of the mechanical compliance of the structure as a function of the crack length and therefore requires a crack propagation controlled experimental test. From a theoretical point of view, this technique comes from an energy balance that needs to be respected during the propagation. Remembering that the energy release rate is G c the energy needed to form a unit surface crack, a propagation of an amount da into a specimen with a thickness B requires an energy E p equal to:

E p=GBda . (5.26)

As a consequence of the propagation, under the action of a fixed external load P, the displacement of the structure will change by a quantity du. The variation of the energy Ee of the external loads produced by the variation of displacement is:

Ee=12

P du . (5.27)

Since the displacement can be written as a function of the load P and of the compliance C, as:

u=CP , (5.28)

by substituting equation (5.28) into (5.27), it yields:

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Ee=12

P2 dC . (5.29)

The global amount of energy remains constant during the fracture process, then:

Ee=E p . (5.30)

By combining equation (5.26) and equation (5.29) and reorganizing the terms, the energy release rate can be written as:

G= 12

P2

BdCda . (5.31)

As aforementioned, the use of the second technique implies the knowledge of the curve C=C a. This curve can be determined from the load vs. displacement plot, that gives C=C P (see Figure 5.31) and from the displacement vs. crack length plot, that gives a=a u .

Figure 5.32. Schematic of a force displacement plot with compliance values in correspondence of different values of crack lengths

In the previous section it was shown that for the structure designed and then tested there is no chance to have a controlled crack propagation even in charge control. If the structure has been pre-cracked, with an initial

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crack length equal to 1.8 µm, the force vs. displacement plot results to be the one displayed in Figure 5.33.With the aid of Figure 5.34, one can imagine to perform an experimental test in charge control. Increasing the value of the charge applied from the null value to Q 1, it is possible to determine all the equilibrium positions of the structure that do not provoke any crack propagation (see Figure 5.27). Once reached the position a, the crack starts propagating and thus the softening branch is followed. If the value of the charge is increased and the new value Q 2 is applied, the new equilibrium position is found in the point b. Thus from the point a to the point c, by increasing of small amounts the value of the applied charge, it is possible to measure all the intermediate loci of the force-displacement curve. From the point c, the electrostatic characteristic (i.e. the dashed line passing through c, that corresponds to the force applied with a value of charge Q 3) does not intersect any other point of the force-displacement curve.

Figure 5.33. Equilibrium position for the structure designed in a virtual experiment carried in charge control

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Figure 5.34. Zoom of the force vs. displacement plot in correspondence of the softening branch

This implies that no other equilibrium position exists in charge control and that the fracture will propagate without any control, causing the break of the specimen.In conclusion, if the real mechanical behavior of the structure will be similar to that obtained with the Finite Element simulation, the use of a pre-cracked structure with an initial crack length less or equal to 1.8 µm, leads to an experimental test that is not completely controllable even in charge control. This means that it will be very difficult to obtain the compliance versus crack length experimental curve.The most reasonable way that could lead to an experimental determination of the stress intensity factor is the first approach described in this section.

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Chapter 6

Numerical representation of polycrystals

6.1. Materials and microstructures

Polysilicon, other materials used for MEMS structural components, ceramic materials in general, but also metals, are usually polycrystalline materials.A polycrystal is an aggregate of crystallites, even called grains, that are monocrystals with a particular ordered lattice and connected each other by a grain boundary (see Figure 6.1).

Figure 6.1. Typical polycrystalline microstructural morphology composed by grains and grain boundaries (from Falk, L.K.L. 2004)

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The microstructural morphology of every material is a consequence of the entire process used to create it. For instance, some important parameters of the process for a polysilicon obtained using a chemical vapor deposition process (CVD) are: the deposition temperature, the growth rate, the pressure used in the reactor and so on. These parameters control, for instance, the final shape of the grains, their average dimensions, the quantity of amorphous phase present in the grain boundary, the presence and size of inclusions and pores, etc.. The ensemble of these properties is often referred to as material microstructure.All physical properties of the material obtained are influenced by every aspect of material microstructure and therefore indirectly from the process parameters adopted in the fabrication.In this work much of the attention is paid in the study of the relationship between microstructure and mechanical properties of polysilicon.The study is carried out performing a number of virtual experiments using Finite element numerical models that simulate the mechanical response of structures made up of polycrystalline silicon materials. The advantage of numerical simulations relies on the fact that they do not require expensive facilities to fabricate the material and time consuming experiments achieve the right calibration of deposition parameters (pressure, temperature, and so on) in order be able to obtain a statistically relevant set of specimens having the desired material microstructures.On the other side, Finite element techniques require a numerical modellization of the polycrystal. To accomplish this, there are mainly two different methodologies.The first one is the automatic image analysis and can be ideally divided into two steps: image acquisition and image processing. Image acquisition needs a tool (optical microscopy imaging, scansion electron microscopy imaging, transmission electron microscopy imaging) for the creation of one or more pictures that show the microstructure of the material to analyze. Image processing is fulfilled with numerical techniques and algorithms that transform the microstructural image in a digital map of grains and grain boundaries.

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The main advantage of this methodology is that it is in principle possible to obtain the most realistic geometrical model of the polycrystal. In some cases, using for instance the backscatter electron microscopy, it is even possible to determine the orientation of every grain of the material scanned with respect to a fixed reference frame. The main drawback is that the technique can be very time consuming if the scanned area or the number of specimen is large or if, as for 3D geometrical microstructural reconstruction, many section images are needed to create the entire geometrical model.The second methodology relies on the use of mathematical tools that are capable to divide a two or three dimensional domain in a set of subdomains with a specified partition rule. These techniques differ each other for the partition rule that allows for the creation of subdomains. Indubitably the most important and used one is the Voronoi tessellation technique [see for instance: Weyer et al., 2002; Mahadevan and Zhao, 2002; Van de Steen et al., 2001; Ghosh et al., 1997], discussed in detail in this chapter.

6.2. Voronoi tessellation as a tool for the creation of numeric polycrystals

Many processes for the fabrication of MEMS materials lead to a creation of a characteristic microstructural morphology: the columnar grain structure. In fact, as it is possible to see in Figure 6.2, every grain can be reasonably considered as a cylinder with the axis corresponding to the growth direction. Besides, it can be added that most of the timens in MEMS structural layers, the thickness of the deposited film is very small if compared to the in-plane size of the structure itself and that in many cases micro-structures are loaded with forces that act principally in the plane of the wafer. These considerations justify the assumption done in this work to consider the geometry of the structure as bi-dimensional. For this reason the technique adopted for the creation of the numerical polycrystal will partition an area and will divide it into subdomains (grains) and frontiers of the domains (grain boundaries).

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Figure 6.2. Columnar morphological structure (from Bastos A. et at., 2006)

6.2.1. Definition and propertiesStarting from a set of n distinct points Q≡{q1 ,q2 , .... , qn}, called Voronoi sites, the Voronoi tessellation is a partitioning technique that divides a space in n cells called tassels.Once defined the Euclidean distance pq as:

pq≡ px2−qx

2 p y2−q y

2 , (6.1)

the i−th tassel is defined as the set of points p≡ px , p y, for which the following relation holds:

pqi pq j ∀ j≠i . (6.2)

The points for which hold the equation:

pqi= pq j , (6.3)

are the edges of the tassel i−th and j−th. Two or more edges can intersect in a point v , called Voronoi vertex (see Figure 6.3).

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Figure 6.3. Voronoi diagram, sites, edges, vertexes

Voronoi tessellation was often used [Mullen et al., 1997; Espinosa and Zavatteri, 2003; Nagaki et al., 1998; Weyer et al., 2002; Mahadevan and Zhao, 2002; Van de Steen et al., 2001; Ghosh et al., 1997] as a tool for the creation and then for the finite element discretization of polycrystals. From Figure 6.3 it is easy to understand the link between Voronoi diagrams and polycrystalline structures: a Voronoi tassel can represent a crystal grain while Voronoi edges represent grain boundaries.One of the most important factors to take into account when performing the Voronoi tessellation with the aim of representing the microstructural morphology is the choice of the position of the sites.Since every site corresponds to a tassel and hence to a grain, the total number of sites will be the same of that of grains. If the size of the area At

fill with tassels is known, together with the average dimension of the grain ⟨Ag ⟩, the number of sites ns that are necessary for the definition of the tessellation is:

ns=At

⟨Ag ⟩ . (6.4)

The overall shape of the microstructural morphology obtained using the Voronoi tessellation depends on the way the sites are chosen. In fact Voronoi sites could be chosen sampling ns points from a generic

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probability distribution function f p= f p x , y .One of the main concern when designing a production process consists in obtaining a deposited layer as homogeneous as possible. In a well engineered process, the microstructural morphology of the layer, as well as the other physical properties, does not have to depend on the position x , y on the wafer.To accomplish this, it is necessary that the microstructural morphology is as much uniform as possible. From a mathematical point of view, the independence of the overall microstructure from the position can be interpreted as an uniform probability distribution to find a site.For this reason a white noise random number generator was used for the sampling of the Voronoi sites.

6.2.2. Centroidal Voronoi tessellationCentroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, and optimal allocation of resources.Given an arbitrary set of site points Z≡{z 1 , z 2 , .... , z k} in ⊂ℜn these points will not in general be the centroids of their associated Voronoi tassels (see Figure 6.4), while for a CVT they will.The use of a centroidal Voronoi tessellations produces a very regular tassels shape. This particular feature can be exploited in order to obtain morphological microstructures with more or less distorted grains.The algorithm developed for the regularization of the grain shape is a recursive algorithm that usually is used to create a CVT starting from a random set of Voronoi sites. The flowchart of the algorithm is illustrated in Table 6.1. The number of regularization passages is fixed at the beginning of the procedure and the random generation is performed once and for all. Then, until the counter i is smaller that the desired number of regularizations, Voronoi tessellation is computed and the centroids corresponding to every tassel are

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Figure 6.4.Non centroidal and centroidal Voronoi tessellation

Table 6.1. Regularization algorithm based on iterative CVT algorithm

calculated. Successively sites' coordinates are substituted with centroids and eventually the tessellation is computed again using the new sites.The main difference between the algorithm developed and the one used for the computation of the CVT, is that the former repeats the cycle for a fixed number of times, while the latter cycles until a norm of the distance

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between the sites and their centroids is less than a fixed value.Figure 6.5 shows the results of the tessellation with a different number of regularizations starting from the same original sites; the bigger the number of regularization is, the less distorted the average grain shape is.

Figure 6.5. Examples of tessellations with a different number of regularizations

6.2.3. Creation of interphasesThe procedure for the statistical representation of grains was thought to be as general as possible. In engineering applications there are many materials that present a distinct phase between two or more grains. In these cases one could imagine that there is a matrix, that can be seen as the 'skeleton' of the material, and aggregates dispersed into it. For most metals and for polycrystals with a small presence of amorphous phase, the matrix disappears and the aggregates, i.e. grains, are linked each other by a grain boundary whose thickness is usually in the order of some fraction of nanometer.

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In other cases, as for instance in concrete, polymers, some metallic alloys, short fiber composites, a geometrical representation of the microstructure needs the presence of the interphase. Figure 6.6 shows the application of the routine for the creation of a material of this kind with a matrix that has a volume fraction respectively equal to 0.3 and 0.5.These materials were not taken into consideration in the present work, but the routine that creates the numerical representation of the statistical creation of the microstructural morphology was designed with the aim to extend the numerical methodology used and described in Chapter 8 to different classes of materials used for MEMS as well for other engineering applications.

Figure 6.6. Example of two microstructures with matrix and aggregates

6.2.4. MEMS structures tessellationThe procedure developed is not capable to perform constrained tessellation. This implies that it is not possible to constrain tassels to lie inside a specified domain with any shape. The result of the tessellation is an area whose shape is more or less rectangular. Thus, in order to create a model that can represent a MEMS structure with grains and grain boundary, it is necessary to develop a procedure that can change the shape of the tessellation obtained, eliminating the grains that lie entirely outside of the domain of the structure and reshaping the ones that lie partially inside and partially outside the structural domain.

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Figure 6.7. Schematic of the procedure for the tessellation of a structure

The procedure developed is schematized in Figure 6.7. Starting from the tessellation and from the structure drawing, through a Cad tool embedded in the Ansys Inc. program it is possible to overlap the two domains. The software performs all the operations of reshaping mentioned previously and then eliminates all the grains that, at the end of the boolean operation, lie outside of the structure. The final result is the structure drawing divided into grains. The last step consists into the discretization of the structure into finite elements. This operation is performed using the automatic mesh generator contained in the preprocessor of Ansys Inc. program that creates the mesh of the grains present into the polycrystal (see Figure 6.8).

Figure 6.8. Example of a discretized microstructure

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6.3. Constitutive models for polysilicon

An important aspect to take into account when modeling a structure using a numerical technique is related to material properties. In the present work polysilicon is modeled as an aggregate of monocrystals. The mechanical behavior of the silicon monocrystal is supposed to be linear elastic until rupture.The linearity assumption can be justified by the experimental results obtained and discussed in Chapter 5, from the papers by Corigliano et al. (2004), where the thick STMicroelectronics polysilicon was tested and by Cacchione et al. (2004) and Cacchione et al. (2005), where the thin STMicroelectronics polysilicon layer was characterized from a mechanical point of view. In all the cases mentioned, polysilicon layers showed a linear behavior.The absence of plastic deformations is due to the high value of dislocation activation energy in the silicon crystal (2.2 eV). Dislocations start moving inside grains causing irreversible plastic deformations only with temperatures higher than about 650°C. Experimentally the absence of plastic deformations was proved loading the test-structures of Chapter 5 up to 95% of the rupture load and subsequently unloading them. In all structures tested no residual deformation or elongation was noticed.

6.3.1. Elastic tensor for cubic crystals

Formulation in the local reference frame Monocrystalline silicon is organized in a cubic face centered (cfc) lattice with a cell side equal to 0.54 nm. Exploiting the symmetries of the crystalline lattice, in a reference frame aligned with three orthogonal sides of the cubic cell, the number of non zero different entries in the stiffness or compliance fourth order elastic tensor reduces from 36 to 3 [see Landau and Lifšits, 1979].

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Figure 6.9. Cubic centered face crystal lattice for silicon

The stiffness elastic tensor can be written as:

S L=[s11 s12 s12 0 0 0s12 s11 s12 0 0 0s12 s12 s11 0 0 00 0 0 s44 0 00 0 0 0 s44 00 0 0 0 0 s44

] , (6.5)

where the superscript L refers to the local reference system. The values of s11, s12 and s 44, used in all numerical simulations are taken from the work published by Brantley (1973):

{s11=165.7 GPas12=63.9 GPas 44=79.6 GPa

. (6.6)

Formulation in the global reference frameNumerical analyses are carried out using a fixed global reference frame. Since the elastic constitutive law of the material is formulated in the local reference frame, it is necessary to transform it from the local to the global frame. To achieve this, a transformation T that operates on second rank tensors is defined. As illustrated in Figure 6.10, T transforms a generic second order tensor, like stress or strain, from the global system to the local one.

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Figure 6.10. Second order tensor transformation from the global to material frame

Having defined the director cosines between the local and global frames as:

cih=cos l ,m {l=1,2 ,3h=x , y , z

, (6.7)

the expression of the stresses in the local frame with respect to the global one (and analogously that one of strains) is:

HK=ij ciH c jK . (6.8)

Using the matricial notation, equation (6.3) can be written as:

L={11

22

33

32

31

12

}=T−1⋅{ xx

yy

zz

zy

zx

xy

}=T−1⋅G . (6.9)

where the superscript G refers to the global reference system and the transformation matrix T, that holds both for stress and strain, is:

T=[c1x

2 c1y2 c1z

2 2c1y c1z 2 c1x c1z 2c1y c1x

c2x2 c2y

2 c2z2 2c2y c2z 2c2x c2z 2 c2y c2x

c3x2 c3y

2 c3z2 2c3y c3z 2 c3x c3z 2c3y c3x

c3x c2x c3y c2y c3z c2z c3y c2zc3z c2y c3z c2xc3x c2z c3x c2yc3y c2x

c3x c1x c3y c1y c3z c1z c3y c1zc3z c1y c3z c1xc3x c1z c3x c1yc3y c1x

c1x c2x c1y c2y c1z c2z c1y c2zc1z c2y c1z c2xc1x c2z c1x c2yc1y c2x

] .

(6.10)

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Since:

{L=T−1G

L=T−1G

L=S LL , (6.11)

from system (6.2) it yields:

S=T S LT−1 . (6.12)

Equation (6.2) allows for the computation of the global stiffness matrix of the silicon polycrystal once assigned the director cosines between global and local reference frame.

Privileged orientationCrystallites orientation is usually casual, but there are some deposition processes [ Sharpe et al., 2001 Tsuchiya et al., 2005;] that cause the layer deposited to have a privileged orientation. This means that a large number of grains presents the same crystallographic direction orthogonal to the plane of the wafer. Privileged orientation, commonly referred to with the noun texture, can influence in some way the mechanical behavior of the material. Therefore the routine developed presents the capability to numerically create a textured layer. As an example, the creation of a [1 0 0] textured layer will be shown.In this case, the local direction 1 corresponds with the global direction z and the director cosines are:

c1={c1x=0c1y=0c1z=1

. (6.13)

Two components of the second direction cosine, for instance c2x and c2y,will be chosen with a random number generator, while the third component c2z is computed in order to satisfy the orthogonality between

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vectors c1 and c2 using the equation:

c2z=−c1x c2xc1y c2y

c1z . (6.14)

The third direction cosines are computed taking the vectorial product:

c3=c1∧c2 . (6.15)

Bi-dimensional approximationThe routine and the Finite Element code written (see Chapter 7) make use of bi-dimensional models. For this reason it is necessary to transform the constitutive law from a general three-dimensional case to a 2D plane stress or plane strain one. In plane strain conditions the stiffness matrix S2D is obtained by taking only the components xx , yy , xy of S. I.e. :

2D=[ xx

yy

xy]=[S xxxx S xxyy S xxxy

S yyxx S yyyy S yyxy

S xyxx S xyyy S xyxy][ xx

yy

xy]=S2D2D . (6.16)

In plane stress the compliance matrix C is first computed inverting S. The plane stress compliance matrix C 2D is then obtained in an analog way to what done for S2D in plane strain, i.e :

2D=[ xx

yy

xy]=[C xxxx C xxyy C xxxy

C yyxx C yyyy C yyxy

C xyxx C xyyy C xyxy][ xx

yy

xy]=C 2D 2D . (6.17)

Inverting C2D

it yields S2D.

6.3.2. Non linear cohesive crack model

General aspectsThe behavior of the material is modeled as linear elastic until a norm of the applied stresses does not exceed a critical level, that can be represented as the nominal fracture resistance, f. When this stress level is reached, a crack is formed into the material. The constitutive law is

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then substituted by a traction vs. displacement jump function. The displacement jump is a norm of the distance between two points lying on the crack surface (in Figure 6.11 P and P') which coincided (in Figure 6.11 P) before the crack creation. The traction can be thought of as the resistance trasnmitted by the material to a further opening of the crack.This is the basic idea first developed by Dugdale (1960) and Barenblatt (1962) originally conceived to eliminate the classical elastic stress field singularity of the linear elastic fracture mechanics and goes under the name of cohesive zone model.This property is perhaps the most important reason for the popularity of cohesive zone models in numerical analyses of fracture [Corigliano, 1993; Allen et al., 1994; Geubelle and Breitenfeld, 1997; Geubelle and Rice, 1995; Ma, 1990; Needleman, 1987; Xu and Needleman, 1996; Yang and Ravi-Chandar, 1996].

Figure 6.11. Shematic of the constitutive law for the sound and fractured material

Choesive models were extensively used for the numerical simulation of the mechanical fracture properties of microstructured materials with different numerical techniques (Zavattieri et al., 2001; Espinosa et al., 2003; Li et al. , 2005; Van de Steen et at., 2002; Sukumar et al., 2003; Ghosh et al., 1997). In all the works cited, with the only exception of that by Sukumar, materials prone to inter-granular fracture were investigated. Intergranular fracture, that is the decohesion of more grains along a

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common grain boundary (see Figure 6.12), is a typical fracture mechanism in ceramic materials.Works carried out by Ballarini (1997) and Chasiotis (2003), showed that the crack path in a polysilicon film can be both transgranular and intergranular. For this reason the software implemented is capable to recognize if the crack is propagating on a grain boundary or into a grain and then to reproduce the two mechanism mentioned and to assign different fracture parameters for grain debonding or cleavage, like for instance different values of the energy release rate, as well as different critical stress values for the crack formation.

Figure 6.12. Typical images of trans-granular (from Celli et al., 2002) and inter-granular (from Moberlychan el at., 1998) fractures

Cohesive zone modelLet us consider a two-dimensional domain with boundary u∪t with u∩t=∅ such that tractions are imposed on t and displacements on u.Let d denote a propagating internal discontinuity (Figure 6.13). In Figure (6.13) n is the unit normal to d, defined by the direction of propagation.

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Figure 6.13. Geometry of the modelled domain and notation

Accordingly, d+ and d

- define the two sides of d acted upon by tractions t+ and t -, which express the interaction in the cohesive zone.The displacement discontinuity u across d can be expressed in terms of the vector u computed on the two sides of the discontinuity, i.e.:

u=u∣d+−u∣d

- . (6.18)

Since in the literature there are no data available regarding the softening behavior of polysilicon and no studies were done in order to check which is the model that better describes the fracture behavior of this material, it was chosen to implement a simple and largely used cohesive law: the one proposed by Camacho and Ortiz in 1996, with some slight modifications in order to take into account the crack closure and interpenetration.Let un and us be the opening and sliding components, respectively, of

the displacement discontinuity vector u=[un u s]T in the local

reference frame of d. Furthermore, let tn and ts be the corresponding normal and tangential traction components of the traction vector t=[ tn t s ]

T. Let tM denote the activation threshold of the undamaged

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material for normal damage initiation (Figure 6.14).Following the formulation proposed by Camacho and Ortiz, a damage mechanism with linear softening branches is assumed. If Gc is defined as the cohesive fracture energy which can be dissipated, a damage variable D is introduced as:

D=1− GGc

, (6.19)

G denoting the cohesive energy left to be dissipated for given opening displacement under the assumption that complete closure of the crack is obtained upon unloading (Figure 6.14), without residual permanent deformations. Let uc denote the limit displacement beyond which the process zone is completely damaged and no tractions can be transmitted.

Figure 6.14. Cohesive interface law.

The negative slope in the softening branch is defined as:

k s=−tM

uc . (6.20)

The effective opening displacement is defined as:

u=⟨un⟩+2c

2us2 , (6.21)

being c a material parameter and interpreted as a coupling coefficient allowing to assign different weights to sliding and opening mechanisms. It is worth noting that if the crack is opening in mode I, the normal

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displacement jump contributes to the effective opening; on the contrary, if the crack is closing, only the tangential component is supposed to participate to the effective displacement jump value. The expression of the cohesive traction is:

t=⟨tn ⟩+2 1

c2 t s

2 , (6.22)

being the components tn and ts defined as:

tn=t⟨un ⟩+ u

k n ⟨un⟩- ; ts=t c2 ut

u . (6.23)

From the expression of the normal component of the traction shown in equation (6.23), it can be noticed that the contact between crack faces istaken into account by the means of a penalty formulation.A normal force, obtained multiplying the amount of interpenetration by a penalty constant k s, is developed in order to eliminate the interpenetration. The contact condition without any interpenetration is obtained if k s∞. Since no experimental friction parameters were available, in order to simplify the model, a frictionless contact was implemented. As a summary, Table 6.2 displays schematically how the cohesive law works.

Figure 6.15. Penalty law for crack faces contact

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Table 6.2. Algorithmic view of the interface law

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Chapter 7

Non linear mechanical simulation algorithms for MEMS

7.1. Overview on MEMS mechanical simulations

MEMS numerical simulations are really challenging. As pointed out in Chapter 2, the most important commercial MEMS systems work exploiting electromagnetic effects that, for instance in gyroscopes, are used to generate driving forces which in turn cause the motion of the system. These devices are immersed in a fluid, that can be either air or an inert gas, with pressure that can be atmospheric or close to vacuum and temperature that can go from thirty, forty degrees below zero, as for space applications, to almost two hundred Celsius degrees, as for automotive applications. In addition to these parameters that can interact changing and influencing the mechanical, static and dynamic response of the system, it must be asserted that the very small dimensions of the system impact dramatically on the physics of the system. As an example, structural weight, that for classical structural engineering has a big importance (as for bridges, dams or big rotating shafts) most of the times can be neglected; damping mechanisms at this scale cannot be modeled as at the macroscale, because the mean free molecular path is, in some cases, of the same order of magnitude of structural components.

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Moreover, MEMS devices are never stand alone products. Micromachined sensors and actuators are placed into bigger systems, like pressure sensors in tires, accelerometers in air–bags, mobile phones or laptops. These systems can undergo accidental loads that, as a consequence, can overload the structure of the sensor and can cause the failure of the MEMS.It is therefore of fundamental importance to develop efficient methodologies for the simulation of the mechanical behavior of microsystems when shocks and overloads occur.For this aim, two main ingredients are thought to be essential in order to obtain accurate results and reasonable CPU simulation time:

• the appropriate description of microstructual morphology, discussed in Chapter 6;

• the use of numerical algorithms and tools for the efficient integration of the equation of motion, discussed in this Chapter.

In the following sections the algorithms and numerical techniques used to achieve the proposed aims are described.

7.2. Direct step by step dynamic analysis

In this section the general formulation for nonlinear structural dynamic problems in the framework of the finite element method is recalled. Time stepping schemes are presented in sections 7.2.1 and 7.2.2. These methods are used for the nonlinear direct analysis of the cases studied in this thesis. When inertia forces cannot be neglected, a fully dynamic analysis must be performed. In the presence of a nonlinear behavior, the analysis is carried out in the time domain, while for the linear case an analysis in the frequency domain is usually preferred.The time interval of interest is subdivided in sequential steps and the following problem is considered: find the approximations for the displacements un1, the velocities vn1 and the accelerations an1 at time instant tn1, given the algorithmic approximations for the same quantities at time tn.

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The semi-discrete momentum equation at the time step tn1 can be written as:

man1FI un1 ,v n1 , tn1−FE tn1=0 , (7.1)

where m is the discretized mass matrix, F I the internal forces vector, that includes damping effects and material elastic or non elastic response, and F E the external forces vector.The time stepping integration algorithms used in this thesis come all from the family of Newmark integration method. For this integrator, having defined t=tn1−tn, the expressions of velocities and displacements are:

un1=un t vnt 2

2 1−2 N anN t2an1 , (7.2)

vn1=vn1−N t anN t an1 . (7.3)

The parameters N and N, called Newmark parameters, control the stability and the artificial viscosity for the linearized system.A wide range of algorithms were developed, in correspondence of different choices of the parameters, but two algorithms are commonly used: they are used here to illustrate two families of different approaches in structural dynamics in sections 7.2.1 and 7.2.2.

7.2.1. Explicit MethodsA typical choice for the Newmark parameters is N=0 and N=0.5 leading to the central difference algorithm (see Hughes, 2001 ). This time integrator is second order accurate, but only conditionally stable. This means that the stability is tied to the request that the time increment t be lower than a critical limit. The stability criterion is obviously a critical component of this scheme, but before discussing this aspect, the equations obtained by substituting the parameter values into equations (7.1)-(7.3) are presented.

an1=m−1[F E tn1−F I un1 ,v n , t n1 ] , (7.4)

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un1=un t v n t2

2 an , (7.5)

vn1=vn t2 a nan1 . (7.6)

The first equation is solved for an1 and it is particularly convenient from a computational point of view when m is a diagonal matrix (therefore, it is common practice to diagonalize it). With these assumptions the vectors an1, vn1, un1 are explicitly dependent on the correspondent vectors at

time tn which are known and it is not necessary to adopt any iterative procedure.The stability request is that the time increment be less than a critical limit t c which is defined by the so called Courant limit:

t c≤2

f max , (7.7)

being f max the highest modal natural frequency in the mesh, which, for the linear case can be estimated conservatively as (see Hughes, 2001):

f max≃2 ch max

, (7.8)

with c and h the sound speed and a characteristic mesh size, respectively. Therefore, an estimate of the time increment is:

t≤ hc

, (7.9)

being a lower-than-one parameter. Equation (7.9) states that the time step cannot exceed the time required by a sound wave to pass through the element in the mesh with the smallest transit time (which for constant c corresponds to the minimum characteristic mesh size). This requirement restricts in the linear case the explicit methods to the simulation of high frequency dominated or wave-like phenomena, since a very large number (order of 106÷107 or even more) of time steps is necessary to solve a

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vibration, low frequency problem. In the latter case, an implicit formulation is more appropriated. When a highly nonlinear behavior is expected, since the time steps can be small, the explicit methods could be again useful, especially when iterative implicit methods could not converge.

7.2.2. Implicit MethodsThe choice N=0.25 and N=0.5 corresponds to the so-called constant acceleration or trapezoidal rule. It is a second order accurate and unconditionally stable integrator for linear problems. This is a typical implicit time integrator currently used in finite element calculations. By substituting the parameter values into equations (7.1)-(7.3) it yields:

man1FI un1 ,v n1 , tn1=FE tn1 , (7.10)

un1=un t v n t2

4 anan1 , (7.11)

vn1=vn t2 a nan1 . (7.12)

By performing some substitution, the equation of motion as a function of the unknowns un1, can be written as:

4 t2 mun1Fn1

I =Fn1E man 4

tvn

4 t2 un . (7.13)

In this case internal forces depend, through un1, on an1. Therefore in case of non linear problems the solution is sought in the framework of a Newton-Raphson iterative technique. A dynamic incremental residual is defined as:

un1 =4

t2 mun1F n1I −Fn1

E −man 4 tv n

4t 2 un . (7.14)

The equation to be solved becomes:

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un1 =0 . (7.15)

The Newton-Raphson technique requires a Taylor expansion of equation (7.15) in iteration j as follows:

un1j1 ≃ un1j [ ∂∂u ]un 1j

un1j1=0 , (7.16)

where

un1j1=un1

j1−un1j . (7.17)

For highly non linear problems this method could lead to a lack of convergence or it could take many iterations before converging. This is the main reason that justifies the use of explicit methods in fracture and contact mechanics.

7.3. Dynamic relaxation algorithm for quasi- static simulations

In this section a technique for non linear static Finite Element analyses is described. It makes use of a dynamic relaxation algorithm [Underwood, 1983; Oakley and Knight, 1995] that basically solves a dynamic problem in which, density and damping parameters are tuned in order to obtain an as fast as possible convergence to a static condition.The algorithm used for the solution of the fracture problem is indeed a static-relaxed dynamic one. Since most fracture mechanics simulations and tests are controllable only using a displacement control, in this section we will make the hypothesis that in a typical fracture simulation the structure is loaded with assigned displacement on a part of the external boundary.The first computation step is accomplished by a sparse direct frontal solver that solves the linear static problem assigning a fixed value of displacement on the constrained boundary. A specific routine, then, computes the value of the prescribed displacement that causes the first

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crack propagation and stores the value of the reaction force corresponding to the assigned displacements. Once the fracture propagates, the problem becomes non linear and the code switches from the static solver to a relaxed dynamic one. This algorithm is based on the central difference method, as explained in paragraph 7.2.1, for the time integration of the equilibrium equations of the dynamic problem.The static solution is obtained by simulating a dynamic event and manipulating properly the values of the mass matrix and damping of the system in order to converge to the static equilibrium as fast as possible.Recalling equation (7.2), for a generic element j, the critical time step can be expressed as:

t cj=

h j

c j . (7.18)

The speed of longitudinal waves in the medium is:

c j= j2 j

j , (7.19)

being j and j the Lamè constants of the material and j the density. Fixing the value of the time increment, by combining equation (7.14) and (7.13), it yields:

j= j2 j th j

2

. (7.20)

Equation (7.12) defines a size-dependent density, that is computed for every element of the discretization.The second parameter that can be tuned to accelerate the static convergence speed is the damping parameter. In this work the damping force used for the dynamic relaxation is expressed as:

f d=mv . (7.21)

The damping force is therefore proportional to a coefficient , to the mass matrix m and to the velocity vector v. The choice of is done in order to

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have a critically damped system, that is expressed as a function of the minimum eigen-frequency of the system min:

≈2min . (7.22)

The stiffness-mass Rayleigh quotient gives an estimate of the minimum eigen-frequency of the system:

min≈wT ktwwT mw

, (7.23)

being w a weight vector, k t the stiffness matrix or a diagonal estimate of the tangent stiffness matrix and m the mass matrix. In this work the weight vector used is the displacement vector u.The algorithm performs the time integration until the static equilibrium is asymptotically reached. Several indicators can be used to verify the convergence of the system: the total kinetic energy norm, the F I−F E residual norm, and so on. The indicator used in this work is the norm of the incremental displacement. Given the displacement vector computed at the time step i-1 and the one computed at the time step i, the equilibrium is reached if:

∥u i−u i−1∥ , (7.24)

being a real positive number with a usual value in the order of 10-6.A typical execution of a quasi-static fracture simulation is sketched in Table 7.1. As already mentioned, the first step allows for the determination of the value of forces and imposed displacements in correspondence of the first crack initiation. Then, for an assigned value of the imposed displacement, the algorithm looks for a static solution integrating the fictitious dynamic problem. Once converged to steady-state, a check for further crack propagation is performed. If the fracture initiation criterion of the material is satisfied in any Gauss point of the system, new cracks are inserted and, leaving unchanged the value of imposed loads, the static equilibrium is sought again. Otherwise, if the fracture initiation criterion is not satisfied, the desired static and

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kinematic quantities are stored, a new value of imposed load or displacements is assigned and the dynamic relaxation is restarted. The algorithm goes on incrementing the applied displacements until the reaction force monitored in the constrained boundary approaches to zero. that corresponds to a complete fracture of the structure. It is worth mentioning that the approach discussed can be applied to both monotonically increasing loads as well as any kind of force history or displacement history applied to the structure.

Table 7.1. Flowchart of the relaxation algorithm

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7.4. Cohesive fracture algorithm

Cohesive fracture algorithms were implemented to solve the static or dynamic fracture propagation using the explicit time integration scheme. The code developed is capable to create cohesive elements, that represent fracture process zones in the body, with a run–time algorithm. This means that at the beginning, when the body is at rest, there are no interface elements present in the model.Since the bulk elements used are 6 noded triangles, interface elements are line element with 6 nodes, as depicted in figure 7.1.

Figure 7.1. 6 nodes triangular elements and interface element

For every integration step, the stress field on all the Gauss points (in light gray in Figure 7.2) of the body are computed. The stress vector n of a generic point lying on a line with a outward unit normal vector n is:

n= n , (7.25)

being the stress tensor in the point considered.The stress vector is then projected in order to compute the normal and tangential components, respectively nn and ns , that are:

nn=nT n , ns=s

T n , (7.26)

A norm of the stress is computed to check if the crack nucleates in P:

= ⟨ nn ⟩+2 ns

2

c2 . (7.27)

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Figure 7.2. Stress vector in a Gauss point and local reference frame

This norm takes into account only tensile normal stress, implying that compressive stresses do not cause the crack nucleation in the material. This simplified model is related to the unknown material properties under compressive loads and to the fact that silicon, as almost all ceramic materials, is supposed to have a much higher resistance in compression than in tension. If one would know the ratio between compression and tension failure strength, he could modify the norm writing for instance:

= 12 ⟨ nn ⟩-

2⟨ nn⟩+

2ns

2

c2 . (7.28)

If the stress norm computed exceeds the nominal fracture resistance f , a new cohesive element is inserted into the model. When a new crack is created into the model, new nodes are created and the element connectivity must be modified. Figure 7.3 shows the topology of a set of elements with an existing pre-crack and how the topology changes once the crack propagates. In this case a crack branching with two branches is shown. The pre-existing nodes are displayed as black circles, while new nodes, created by the crack propagation, are displayed as cyan squares.The cohesive interfaces contribution is taken into account in the internal force vector F I . This vector can be split into a damping term, an elastic term and a cohesive term:

F I un ,vn =dmd k vnk unF cI un . (7.29)

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Figure 7.3. Topology change due to crack propagation (gray squares: the new nodes)

The equation of motion to solve for is thus:

man1=FE−F I , (7.30)

that allows for the computation of the accelerations and hence of the velocities and displacements. Cohesive tractions and shears components are computed on every interface element making use of the model described in Chapter 5.Referring to the picture of Figure 7.4, from the values of tn and t s computed on the line with the length i that ideally represents the interface, the nodal forces in the local reference frame of the i-th interface are computed by:

F in=h∫

i[ N T

−N T]tn s' ds ' , F is=h∫

i[ N T

−N T]ts s ' ds ' , (7.31)

where h is the out of plane thickness of the interface and N the shape functions of the interface element. The interface nodal forces are subsequently transformed from the local reference frame to the global one (see Figure 7.4):

[F ix

F iy]=[ cos sin

−sin cos ][F inF it ] . (7.32)

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This computation is performed on every interface in order to assemble the vector F c

I .

Figure 7.4. Example of sliding tractions on the interface

7.5. Pure explicit algorithms for micro-scale simulations: drawbacks

Computational explicit analyses of MEMS at the micro-scale are very CPU time consuming for two main reasons.Usually MEMS are placed into systems that are much bigger than the MEMS itself. For instance the characteristic size of a MEMS structure is in the order of 100 µm, while the devices they are embedded into, like laptop computers or automotive components, are several tenth of centimeters. The typical duration of a mechanical shock in one of these components is a fraction of millisecond. If one would carry a numerical simulation of the fracture propagation in a MEMS attached to a part that undergoes shock or impact, he should discretize the structure using an element size smaller than the dimension of the process zone in the cracked region. Assuming to have a polysilicon with a critical stress intensity factor equal to 2 MPam and a nominal tensile resistance f =2 GPa , the radius of the cohesive zone is [Camacho and Ortiz, 1996]:

Rc=8 K IC

f 2

≃0.4 m . (7.33)

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Making the hypothesis that the characteristic element size is then 0.1 µm, since the speed of sound in the material is about:

c≃ E= 150109

2330=8023m / s , (7.34)

the critical time step of an explicit simulation is:

t c=hc=1.2510−11 s . (7.35)

This means that the integration of the entire shock event would need about 108 steps.To this, it has to be added that the tessellation algorithm, most of the times, produces very short grain boundary segments (see Figure 7.5). As a consequence, the automatic mesh generator, that collocates at least one element on the segment, will produce either very small elements or very distorted ones, as illustrated in the detail displayed in Figure 7.5. Thus, the critical time step will be influenced by these distorted elements, whose characteristic length (for instance the radius of the circle inscribed into the element) is by far smaller than the one of non distorted elements. The resulting critical time step is usually in the order of 10-12.Therefore it appears evident that the time step that can be adopted in an explicit analysis is a limiting factor for the use of explicit integration for dynamic non-linear simulations at this scale.To reduce the CPU time and to make dynamic fracture simulations feasible, two different techniques were developed. The first one consists in the use of a mixed time integrator, while the second one is a multi-time-step integration algorithm. These two techniques are described in detail in the next sections.

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Figure 7.5. Tessellated structure and detail with distorted elements

7.6. Implicit–explicit integration scheme

The mixed time integrator consists in the use of an implicit integrator until the material remains elastic and a switch to the explicit integrator when fracture occurs.The implicit integration exploits the linearity of the dynamic system. If this condition holds, the integrator, that is the one also referred as “trapezoidal rule” Newmark algorithm, is unconditionally stable. The time step size is therefore chosen in order to have an accurate solution. The methodology developed is schematically illustrated in the flowchart of Table 7.2. As a first step, the computation of the explicit time step is performed. Then the implicit time step is fixed. Its value is chosen as a multiple of the explicit analysis time step. Usually the ratio between implicit and explicit time step is in the order of 105, that leads to a time step of about 10-7 s. At the end of every integration step, the stress norm is computed on all the Gauss points of the model. The implicit algorithm continues the integration until the solid remains linear elastic. When the first crack initiation occurs, one step back is performed and the instant of the initiation is computed.

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Table 7.2. Schematic flowchart of the implicit-explicit algorithm

Assuming to be n the maximum value of the stress norm at the instant tn , when no fractures are present in the body, and n1 the maximum

value of the stress norm at the instant tn1 , when the fracture initiation criterion is somewhere satisfied (i.e. n1 f ), and that the stress history from tn to tn1 is linear, one can write:

= nn1−n

tn1−tnt n1−t . (7.36)

The instant of initiation is computed by replacing, in equation (7.36), with the maximum admissible stress f . After some manipulations, it is:

t f=t n1− f −n

n1−nt n1−tn . (7.37)

One single implicit integration step is then performed from tn to t f .

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The algorithm then switches from the implicit to the explicit form and goes on until the end of the analysis.

7.7. FETI algorithms for multi-domain problems

In this section a very versatile algorithm for the integration of dynamic non linear problems is discussed. This algorithm was originally formulated by Gravouil and Comberscure in 2001 to solve the dynamic problem of a complex structure by dividing it into subdomains and integrating every sub-domain with a different time integrator, making use of different time steps. In the literature there is a variety of these algorithms, often called subcycling algorithms [see for instance Smolinski at al., 1996; Wu and Smolinsky, 2001; Daniel 2003 and 2001; Neal and Belytschko, 1989;]. Some of them are not stable even if every domain is integrated using a time step smaller than the critical one, while others achieve stability but they sacrifice accuracy, since on the interface between two domains the momentum conservation is not respected. The algorithm used here was chosen because it can be proved that is always stable (see the original work for the proof) under the condition that the domains are independently integrated using a time step smaller than the critical one. The price to pay for stability is a small energy dissipation on the interface that connects the domains. Authors proved that this quantity is often in the oder of some part per thousand of the total energy of the system.The algorithm is used in this thesis with a main difference from the original formulation. These topics, as well as the original formulation, will be discussed in the next paragraphs.

7.7.1. Comberscure-Gravouil algorithmLet us consider a domain with prescribed displacements on the boundary u and prescribed tractions on the remaining boundary t . Using a finite element discretization of the continuous problem in space and a numerical time stepping integrator, the discretized equations for

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the system can be written as :

man1FI un1 ,v n1=F

E t n1 , ∀ t ∈ [0,T ] . (7.38)

u0=u∣t=0 , v0=v∣t=0 . (7.39)

un1=ud tn1 on u ∀ tn1 ∈ [0, T ] . (7.40)

m denotes the mass matrix, an1 , v n1 , un1 the nodal accelerations, velocities and displacements at time tn1 . F I and F E , are respectively the internal and external nodal forces; u0 and v 0 are the specified initial displacement and velocity vectors and ud denotes the displacement boundary condition on u at time tn1 . From the global problem in space, one can perform a decomposition of the structure into s substructures. The dynamic equations are solved in each subdomain k:

m k a k F I k u ,v =F E k F L k , ∀ k ∈ [1,... , s] , (7.41)

being F Lk the 'link' force acting on the domain k that takes into account the interactions between different domains. The consistency of the global problem is enforced by conditions of continuity at the interfaces between different domains:

∑k=1

p

C k T v k =0 , (7.42)

while the connecting forces at the interface between subdomains are:

F Lk=C k T . (7.43)

C k are the Boolean connectivity matrices and the vector of Lagrange multipliers.The solution of the dynamic problem with the imposition of continuity of velocities on the boundary using Lagrange multipliers can be obtained by the minimization of a functional in which the work associated with the multipliers is introduced.

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∫0

t

∑i=k

s

[12vk T mk v k −F k E−F k I T u k −C k v]dt=0. (7.44)

Let us assume to have two domains: 1, associated with a coarse time step t1=a t2 and 2, associated with a fine time scale t2 ,as illustrated in

Figure (7.6).The minimization of the functional and the subsequent discretization in space and time, leads to the set of algebraic equations:

m1an1=Fn

E1−F I 1 un ,v nC 1T n , (7.45)

m2a j2=F j

E2 −F I 2 u j , v jC 2T j , (7.46)

with

C 1v j1C 2 v j

2=0 . (7.47)

The continuity of velocities expressed by equation (7.47) on the fine scale requires that the velocity at the edge of the subdomain be evaluated attime t j . Similarly, equations (7.45) and (7.46) are coupled through the Lagrange multipliers and require that the dual quantities of the edge velocities, i.e. the Lagrange multipliers on the fine time scale, be evaluated.

Figure 7.6. Time steps of domains 1 and 2

For this purpose, the following transition operators from the fine time scale to the coarse time scale (along the edges of the subdomains) were proposed by the Authors of the original work:

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C 1v j1=1− j

a C 1 vm1 j

aC 1 vm−1

1 . (7.48)

The system of equations (7.45)-(7.47) can be solved by dividing all unknowns of the problem into a 'constrained' group (indicated with the superscript F) and 'unconstrained' group (indicated with the superscript L). The first category corresponds to the solution of the equilibrium subdomain by subdomain without taking into account the interface forces. The second category corresponds to correcting terms for the interface forces between subdomains. Introducing the central difference scheme, the unknowns become:

{ un1F =un

un1L =0

{ v n1F = v n1

F 12 t an1

F

vn1L =1

2 t an1

L , (7.49)

being v n1 the velocity predictor at time tn1 . The problem (7.44), split it into unconstrained problem and constrained problem is:

[m1 00 m2][ am

1F

a j2F ]=[F m

E1−F m

I 1

F jE2

−F jI 2 ] , (7.50)

[ am1 0 −aC 1T

0 m2 −C 2T

−aC 1 −C 2 0 ][ am1L

a j2L

]=[ 00

2t C

1 v j1F C 2 v j

2F ] . (7.51)

The first and second equation of (7.51) can be inserted in the third set in order to have a problem condensed at the interfaces on the fine time scale:

H j=−C 1 v j1F C 2 v j

2F , (7.52)

Being H:

H= t2

aC 1m1−1C 1TC 2m2−1

C 2T . (7.53)

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the interface energy term T C 1v1C 2 v2 is either zero (multi-domain coupling with a single time step) or positive (different time steps). This means that some energy may be dissipated because of linear interpolations of velocities. In any case, the stability of the explicit integration of the system is not affected by the coupling. It is always ensured by the verification of the Courant condition in each subdomain. Table 7.3 shows the flowchart of a complete integration time step of the two domains. Starting from the knowledge of the state of the system at time tn , the unconstrained problem on the domain 1 (with the coarse time scale) is solved, obtaining un1

1 , v n11F and an1

1F . Then, with a loop

on the a time steps: the problem on the unconstrained domain is solved; the interpolated velocities v j

1 on the interface are computed using (7.48);

the Lagrange multipliers are computed using (7.52); the constrained quantities on the fine scale domain are obtained using the second set of equations in (7.51). At the end of the loop using the first set of equations in (7.51), it is possible to compute the constrained quantities on the coarse time scale domain.It is worth noting that the formulation was done without using the additional hypothesis of linearity of the problem. The method can be applied to all categories of material behaviors, even in small or finite displacement and deformations.

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Table 7.3. Flowchart of the integration of a complete time step with the Combescure and Gravouil algorithm

7.7.2. Application of the domain decomposition algorithm to multi-connected domains with crack propagation

As discussed in Section 7.5, one of the main problems when performing micro-scale simulations is the presence of very distorted elements that lead to very small time step in explicit integration schemes. In this work, the Gravouil-Combescure domain decomposition method was used to bypass this problem and increase the efficiency of the time integration routines. The Finite elements discretization is divided into two subdomains, with two different time scales. The former contains the distorted elements, and it is associated with a very fine time step; the latter contains non distorted elements and is integrated using a bigger time step. The distortion of an element is defined considering the size of

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the radius of the circle inscribed into the element, that can be considered as the characteristic length of the element. Elements with a small value of the radius are associated with a small time step. For a tasseled structure, it is possible to notice that there are some elements, the distorted ones, with a value of the radius that can be one order of magnitude less than the mean value (see Figure 7.7).

Figure 7.7. Typical incenter radii distribution for a tesselled structure.

If a classical explicit integration scheme is used, the time step will be reduced almost by a factor 10 due to the presence of distorted elements.Therefore, using the domain decomposition algorithm, it is possible to use the small time step for integration of the few distorted elements present in the mesh, while a bigger time step is used to integrate all the other elements of the discretization.The domain is partitioned once fixed r t , the threshold value of the radius. If, for a generic element e, the relation r er t holds, being r e the radius of the element, the element is inserted into the fine time step subdomain. Otherwise it is inserted in the coarse time step element set.The typical result of the partition procedure is displayed in Figure 7.8.

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Figure 7.8. Partition of a domain into two sets (respectively in white and black)

The value of the threshold is computed using the formula:

r t=r−a r , (7.54)

where r is the mean value of the radii, r the standard deviation of radii distribution and a a real positive number. Different simulations were performed changing the value of the parameter a and it was found that using the value a=2 smaller CPU times were often achieved.Since the original algorithm was not explicitly formulated for multi-connected domain, as that obtained by the partitioning scheme adopted, some tests were carried out to understand the grade of accuracy of this method for micro-scale dynamic simulations.The implementation of the algorithm in a Finite element code conceived for the dynamic crack propagation is straightforward. When the fracture propagates starting from a Gauss point located in an internal position of the domain 1 or 2, the standard procedure for the insertion of cohesive elements is followed: new nodes are created in the domain of interest and the element connectivity is modified in order to produce a real discontinuity in the body. A particular condition to care about is the case in which the crack would propagate on the interface between the domains. In this case, on one side one should need the presence of the discontinuity to represent the crack, while on the other, using the Gravouil-Combescure algorithm, one should enforce the velocity continuity for every integration time step, in contradiction with the previous request. The first approach applied to solve this problem

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consisted in: checking for every time step possible further fracture propagations or nucleations; move all the bulk triangular elements with a face crossed by a crack in the fine time step domain; redefine the interface between elements and then perform time integration. With this procedure, some of the new nodes present on the interface were originally internal nodes of the domain 1 or 2. This procedure, even though very appealing from a computational point of view (because it allowed to move all non-linear elements in the fine time scale domain), produced unstable results.A different strategy to solve the problem was then adopted. It consists in preventing the crack propagation on the time step interface. This assumption provokes a limitation in the choice of the crack path, that can not run through the domains interface. Since the number of distorted elements in the mesh is very limited, this additional hypothesis is not thought to condition the quality of the overall solution of the problem.

Plane wave propagationIn order to check the accuracy and compare the numerical solution with an analytical one, a very simple case of linear elastic wave propagation was investigated. The problem solved is that of a bar perfectly clamped on one extremity and loaded with an uniform pressure p0 on the free edge. Using a value for the Poisson's ratio equal to zero, the problem reduces to a one-dimensional case, in which the application of the load causes a tensile stress wave to propagate in the bar with a velocity c=E / :

c= E = 200106

7800=5063.7 m/s , (7.55)

being E the Young modulus and the mass density of the material. The value of the stress on the wave front is p0=1 MPa until the clamped edge is not reached. Since the constraint is ideally perfect, the wave is totally reflected and the wave is bounced back and the new value on the propagating front results to be 2 p0 .

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The domain is a 10 by 4 mm2 rectangular area discretized with 3155 elements. The geometry of the structure was obtained with a Voronoi tessellation, whose discretization created distorted elements. Every tassel is supposed to be linear elastic and isotropic in order to have an homogeneous material. The domain partition is the one shown in Figure 7.8, the tessellation and the structural scheme are depicted in Figure 7.9.

Figure 7.9. Tessellated area and structural scheme for the plane wave propagation simulation.

In the simulation, the stress value was monitored at three points: a point in the fine time step domain (upper plot in Figure 7.10); a point on the coarse time step domain (central plot in Figure 7.10); a point on the interface between the two domains (lower plot in Figure 7.10).Figure 7.10 shows the comparison of the stress in the direction of propagation obtained with the multi-domain algorithm (continuous line) with the analytical solution (dashed line). The agreement is excellent in the three cases analyzed. It is remarkable that the solution obtained with the multi-domain algorithm was 13 times faster than the non partitioned explicit algorithm.

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Figure 7.10. Stress plot in the propagation direction for: a fine time step domain node (upper plot); a coarse time step domain node (central plot); interface node (lower plot).

Dynamic crack propagationThe test performed to asses the quality of the procedure developed for dynamic crack propagation was a tensile test of a 'dog-bone' specimen loaded with a monotonically growing tensile load. The geometry of the specimen, its dimensions, and the discretization used (the black elements are the distorted ones) are displayed in Figure 7.11. The total number of elements is 2163 and the number of elements belonging to the fine time scale domain is 103. The time step ratio between the two domains is 10, being the smaller one equal to 2.56 10-10 s.

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Figure 7.11. Drawing of the specimen, structural scheme and mesh (distorted elements in black).

Figure 7.12. Applied pressure vs. time plot

The force is applied as a uniform pressure acting on the top surface and it varies linearly from 0 to 150 MPa (Figure 7.12). The analysis time interval is 6 µs. The values of the mean displacement in the vertical direction of the top surface and the total length of broken interfaces (the ones with a

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damage value equal to the unity) were monitored. The simulation was carried with and without the subdomain partitioning technique and the results were compared. From Figure 7.13 it is possible to notice that the displacements of the loaded surface are almost exactly the same for both the cases analyzed. The total amount of the cracked surfaces is very similar with the two different algorithm used, confirming the quality of the approach proposed. It is worth noting that the evolution in the time of the crack path is somehow 'faster' for the pure central difference algorithm. This difference can be related to the fact that, being the integration time step in this case ten times smaller, some difference in the solution for highly non linear problems (like fracture propagation) is to be expected. This difference is reflected in the images showing the fracture path (Figure 7.14), that is very similar, but not exactly identical.

Figure 7.13. Results of he simulation carried using domain decomposition algorithm (squares) and without using it (line)

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Figure 7.14. Crack paths obtained with (left) and without (right) the multi-domain algorithmm

7.8. Mass scaling and final remarks

A classical technique that is often used in explicit dynamics to speed up the time marching solution is the mass scaling. This method is based on the idea to change the density of the material in order to increase the value of the critical time step. Recalling equation (7.9) and expressing the velocity of a propagating wave as c=K / , being K the bulk modulus of the material, the critical time step is:

t≤h K

. (7.56)

If one changes the density of the material by multiplying it by a factor , the new value of the critical time step is:

t= t . (7.57)

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If is greater than one, the critical time step increases and thus the number of time steps necessary for the integration of the problem decreases, obtaining a faster solution.Mass scaling (or pumping) has to be used very carefully. In fact, by modifying the value of the density, the entire dynamic of the system is altered. Generally speaking this method can be successfully used when the inertial term in the equilibrium equations is small if compared for instance to elastic or viscous contributions, that happens when masses are almost negligible or when accelerations are small, and no rate effects are present.In the code developed there is the possibility to scale the mass of the system, but this technique is not used as default, since in fracture propagation problems the change of density implies a change in the fracture speed, that have a big impact on overall results.As a closing remark, Figure 7.15 shows the CPU time necessary for solution of a generic micro-scale fracture dynamics problem on the same machine. It is possible to notice, that the implicit-explicit algorithm allows for a five times faster solution with respect to the pure explicit algorithm. The use of the multi-domains algorithm speeds up the solver more than ten times, reducing the total computational cost of more than seventy times the original one. Then, if the hypotheses for its application hold, the mass scaling (in the example of Figure 7.15 =1000 was used) allows for an even faster solution.As discussed in detail, the CPU time is a key issue for this kind of problems. To give an idea, the prevision for the execution time of a simulation with about 3000 elements with an analysis time of 0.5 ms on a IntelTM PentiumTM 4 processor with 512 Mb of Ram memory with a pure explicit solver is about 4 weeks.

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Figure 7.15. CPU time evolution

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Chapter 8

Parametric study of polycrystalline silicon

8.1. Methodology adopted

In the present Chapter a parametric numerical study is carried out in order to understand the role of some important microstructural parameters, like grain average size and morphology, in the determination of the mechanical properties of polysilicon. The study is performed using the Voronoi tessellation technique discussed in Chapter 6 and the dynamic relaxation algorithm, described in Chapter 7.For every assigned value of the parameter taken into consideration, a number of simulations is performed. Every simulation is carried on nominally identical specimens that are produced with a different tessellation runs and therefore present a different microstructure and a different grain orientation. This is done with the aim to simulate a real experimental campaign, in which every specimen is slightly different from another. The data obtained are then interpreted by assuming statistical probability distributions, like the Gaussian or the Weibull one.

8.2. Young's modulus evaluation and prediction

The first mechanical property to be studied was the Young's modulus of the material. As pointed out in Chapter 5, in the scientific literature there is still a large dispersion of the data regarding the Young's modulus of

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polysilicon. This scatter could be due not only to the different measurement techniques, but due, for instance, to the different processes used to create the material. In this thesis two factors were investigated in order to understand their influence on the properties of the homogeneized material: the grain size and the grain texture.To accomplish this, a l=10 m by w=3 m rectangular shaped pure tension specimen was used. With reference to Figure 8.1 a uniform tensile surface load p was applied on the top surface and the mean value of the displacement on the extremity l was measured. Since:

E= F lA l

= p AlA l

= p ll

, (8.1)

it was possible to measure the value of the Young's modulus.The simulations were carried out with the hypothesis of plane stress conditions assuming an out of plane thickness equal to the unity.

Figure 8.1. Tensile test for the Young's modulus measure schematic

8.2.1. Grain size effectThe first parameter investigated was the average grain size. Hereafter the grain size will be defined as the size of the radius of a circle with the same average area as the grains. The values of the grain sizes considered varied in the interval 0.5÷2.3 µm with a difference of 0.2 µm from a set to another. For every set a number of 500 linear elastic simulations was performed. No privileged orientation was used for this scope. The first result displayed in Figure 8.2 shows how the data obtained are clearly

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well represented by a normal distribution function, usually encountered in experimental tests.The second important aspect to notice is displayed in Figure 8.3. Having defined the COV (coefficient of variation) as:

COV x =100 x

⟨ x ⟩ , (8.2)

being ⟨ x ⟩ the mean value of the variable x and x its standard deviation, the plot shows how, decreasing the average grain size, the measured scatter of Young's modulus decreases. Besides, it is interesting to underline how the absolute values of the coefficient of variation are very small. For a fine grained material the standard deviation is in the order of 1% of Young's modulus value, while for a very coarse grained material (or for a specimen whose dimension are comparable with that of the grains) this value barely reaches 5%. These results can be surprising, but there are experimental evidences of this (see for example paragraph 5.2, Cacchione et al., 2006, or Cacchione et al. 2004 where a very small specimens compared to the grain size were tested and a small dispersion around the mean value of Young modulus was measured).

Figure 8.2. Statistical distribution of the Young's modulus obtained with numerical simulations

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Figure 8.3. Coefficient of variation of the Young modulus vs the average grain number and grain size.

8.1.1. Texture effectThe second parameter that is thought to have an important role in the determination of Young's modulus is the texture, defined as a privileged orientation of the grains. Four different conditions were analyzed by carrying out 500 simulations for each texture. The texture analyzed are characterized by the fact that they present a privileged specific crystal orientation aligned with the out of plane normal direction. In this study the following cases were compared: ⟨100 ⟩ , ⟨110⟩ , ⟨111 ⟩ and no texture at all, i.e. random orientation. The grain size used in these simulations was 0.5 µm.The results are condensed in table 8.1, where for each material the mean Young's modulus and the coefficient of variation are reported. The results of the simulations were compared with the experimental works done for the elastic characterization of polysilicon available in the scientific literature that reported some indications regarding the texture of the material tested. The work of Tsuchiya and coworkers, published in 2005, reports a measured value for a ⟨111 ⟩ polysilicon equal to 168.9 GPa. This value is surprisingly near to the 170.0 GPa obtained with the numerical

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Table 8.1: Young's moduli obtained with numerical simulations adopting different textures

simulations. The second work found is the one by Sharpe and coworkers, published in 2001, that reports a value of the modulus equal to 158 GPa for a ⟨110⟩ oriented polysilicon. Again the value obtained with numerical simulations, 162.5 GPa, is very close to that value.Let us now consider the thick layer of polysilicon produced with the ThELMATM process described in Chapter 5. Differently from many other MEMS processes, whose thickness is seldom bigger than 2.5 µm, ThELMATM structural layer is 15 µm thick. The production process of the layer starts with the deposition of polysilicon germs. These germs are tiny silicon crystals deposited on the wafer. In the reactor the germs grow and give rise to the columnar morphological structure. Crystalline growth is 'competitive' in the sense that crystals with a better orientation with respect to the growth direction can grow, while others, that present a worse orientation, can not.

Figure 8.4. Detail of the microstructural columnar morphology

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This fact can be better understood with the aid of Figure 8.4. In the lower part of the figure it is possible to see a considerable number of grains that stop their growth, while others continue the growth from the bottom to the top of the layer. Since usually the privileged growth orientation coincide with the crystallographic axis, it is reasonable to assume that many grains will present a crystallographic axis aligned with the growth direction, i.e. the normal direction to the plane of the wafer. Since the silicon cell is cubic, one can assume that the crystallographic privileged orientation, for a layer thick enough to present a columnar structure created with a competitive growth, could be the ⟨100 ⟩ , that corresponds to one of the sides of the cube.The value of Young's modulus of the material obtained measuring the structure discussed in paragraph 5.3.3 is 143 GPa, while the results obtained using a texture ⟨100 ⟩ in the numerical simulations is 138.4 GPa, very close to the measured value. This could be interpreted in a certain sense as a very rough identification procedure that confirms the assumptions done about the texture of the material. Since other parameters could influence the final value of Young's modulus, an experimental measure of the texture should be performed to confirm what asserted.

8.3. Fracture properties

The same approach followed for elastic simulations was used to understand the role of some microstructural parameters on fracture properties. Since polysilicon is a very brittle material, its fracture properties are often evaluated in the framework of Weibull statistics, discussed in Chapter 4. This implies that it is necessary to simulate the behavior of many nominally identical specimens for every assigned set of parameters. The results are then interpreted as coming from different tests carried out on a large number of identical structures and interpreted with the Weibull approach. The tests were realized on a pure tension specimen with a l=5 m by w=2 m rectangular shape in plane stress conditions. Except for the study of the influence of the texture on fracture

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properties, and the investigation on the role of the specimen size on the overall fracture resistance, for the other tests executed, three different categories of polysilicons were considered:

• one with the same strength properties on grains and grain boundaries (Material I);

• one with grain boundaries weaker than grains (Material II);• one with grains weaker than grains boundaries (Material III).

The critical energy release rate used in all the simulations was Gc=7 N /m , that, using the typical values of Young's modulus for

polysilicon ( E=160 GPa , =0.2 ), corresponds to a K IC≈1.1MPam , a common value found in literature.The result of every simulation is a force versus displacement plot. With this curve it is then possible to compute the maximum nominal stress cr

that acted during the test:

cr=max F

A . (8.3)

From an entire simulation set a vector containing the maximum stress of every run is obtained. From this vector it is possible to compute the cumulated failure probability. The cumulated probability versus nominal stress plot is then fitted with the Weibull probability function:

P f =1−exp[− 0 m] , (8.4)

and the Weibull parameters m and 0 are then determined.The number of simulations for every set was chosen to be 50. This in order to have a statistical meaningful sample and because usually experimental measures done for the mechanical characterization of MEMS materials on microstructures or on micro-specimens are in the order of 30–60.

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Figure 8.5. Sound and fractured specimens examples for: Material I (left), Material II (center) and Material III (right)

8.3.1. Grain orientation effectThe first investigation regarded the role of the texture on the fracture properties. The different orientations of two or more grains can be a cause of stress concentrations on the interface between grains. This is due to the fact that elastic tensor of the grains expressed in the global reference frame can differ from a grain to another because of the random orientation, while compatibility on the grain boundary must be respected. Generally this condition leads to a stress concentration that can cause a premature failure with respect to an homogeneous material. The texture of a polycrystalline film is the privileged orientation of its crystallites. As a consequence texture can be seen like a constraint imposed to the grains on its possible spacial orientations and therefore as a reduction of the dimension of the space of the possible misalignment between different grains. Therefore from this point of view one should expect that a textured film, with a minor grade of misalignment between grains, should be less affected from grain boundary stress concentrations and therefore its overall fracture performance should be better than a non textured film.As it is possible to appreciate in Figure 8.6 and from Table 8.2, numerical simulations confirmed the worst performance of non textured film with respect to textured ones. Between the textured films, it is remarkable that the ⟨110⟩ oriented film appears to be the most performing one because it

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Figure 8.6. Cumulative failure plots for different textured films

Table 8.2. Weibull parameters for differently textured films

has the highest values of the Weibull modulus m and threshold stress 0 .

8.3.2. Specimen size effectAs reported in Chapter 4, polysilicon films fracture properties manifest a strong dependence on the volume of the specimen used to carry the experimental test [Ding et al. (2001), Jadaan et al. (2003)]. Weibull statistics was proved to be very accurate in predicting the failure probability of the structure as a function of its volume. If two structures respectively with volume V and V , made of the same material (and thus with the same Weibull modulus m and the same representative volume V r ) are considered, the scaling law for the Weibull threshold

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stress 0 is:

0V= 0

V 1

1m , (8.5)

where 0 V is the threshold stress of the structure with volume V and

0V the threshold stress of the structure with volume V . Simulations

were executed in order to understand if the numerical procedure that has been set up is capable to catch volume depending effects on the mechanical properties of the material and if, in the case of a positive answer, results obtained would be in agreement with Weibull statistics.For this purpose, three specimens with unit thickness were used. Their size is respectively: 5 µm x 2 µm, 7.071 µm x 2.828 µm and 8.660 µm x 3.464 µm. The volume ratio , defined as the ratio of the specimen volume over the volume of the smaller specimen, is: 1, 2, 3.Being the Weibull modulus obtained from the data reduction for the different specimens very similar in the three cases, the mean value, corresponding to 78, was used for the prediction of the threshold stress when applying equation (8.5). Figure 8.7 displays the results obtained and the interpolated curves resulting from the application of the Weibull distribution. Table 8.3 shows how the numerical values of 0 obtained are in very good agreement with the predictions done using the reference 0 value of the smaller structure to forecast the 0

V Weibull parameters for the bigger ones. The difference between the predicted and the reduced values for the cases with =2 and =3 are below the 1%.

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Figure 8.7. Cumulative failure probability plots for different specimen sizes

Table 8.3. Simulated Weibull threshold stress and prediction using the Weibull scaling law

8.1.2. Grain size effectIn the Material Science literature it is a common opinion that grain size plays a very important role in determining fracture properties of ceramic materials [see e.g. Lawn, 1993]. For this reason, the effect of the grain size on polysilicon film was investigated. Three different material fracture behaviors, as explained in section 8.3, were taken into account. For every set with a specific material a subset of 50 simulations with different grain sizes was created. The grain sizes investigated were 0.2 / 0.3 / 0.4 / 0.55 / 0.75 µm. The specimen size is 5 µm x 2 µm and the grain orientation is random. The typical microstructural morphology obtained for the

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Figure 8.8. Microstructural morphology of specimens with different grain sizes

different sets is depicted in Figure 8.8.In Table 8.4 the nominal strength measured on the numerical tensile tests are shown at varying average grain size and fracture activation properties of polysilicon. From table 8.4 it is possible to notice that for every material the value of the threshold stress remains almost constant (with variation of less than 1% of the mean value going from one grain size to another) showing an independence from the grain size. Moreover, the results of the simulations confirm the brittleness of the material. In fact the values of the threshold stress are close to the opening stress of the weakest interface.This result is in agreement with the use of Weibull theory for the evaluation of material’s strength, also called the ‘weakest link approach’. Another interesting remark can be done concerning the ratio of the average value of 0 and t lower

max , where t lowermax is the lower value of interface

resistance in the model. For the material with equal grain and grain boundary this value is 0.9333; for the material with the grain stronger than the grain boundary is 0.9485 and for the last one is 0.9749. Interpreting this ratio as a performance index of the material, one should conclude that the material with weak grain boundaries appears to be the best one. The reason of this could be partially explained as follows. When the grain is weaker than the grain boundary, the crack starts from a Finite

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element edge located inside the grain; vice-versa, when the grain boundary is weaker than the grain, the first opened interface is to be sought into the grain boundary. Since the total number of edges on the grain boundary is small compared to the total number of edges into the grains, the probability to find a broken interface, and thus a crack, into the grain is bigger than the one of finding it onto the grain boundary. It turns out that it will be easier for a material of the first class to break when the mean stress is close to the critical one for the crack propagation.

Table 8.4. Threshold stress results.

Table 8.5 shows the variation of the Weibull modulus m with respect to the grain size. In this case the link between the grain size and the modulus appears to be evident. For every material, as the grains become smaller, there is a trend toward an increase of modulus m, i.e. toward a reduction of the spread of the interval in which there is a sharp variation in the fracture probability. This effect is intuitively understandable. In a specimen with a small number of grains the overall mechanical behavior is strongly influenced by the shape and orientation of its grains. As the number of grains increases, the influence of a single grain onto the global behavior becomes less important and, if the number of grains is very large, the global response of the system should converge to a mean value. This is the reason why as the grain size diminishes, the Weibull’s

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modulus increases. The second aspect that it is worth noticing is the comparison of the modulus for the three material models analyzed. The material with the highest values of the modulus m is the one with trans-granular privileged crack propagation, while the material with an intra-granular crack path appears to be the ‘less’ reliable, because its modulus is always smaller than in the previous case. Finally, the material without any privileged path seems to exhibit an intermediate behavior.

Table 8.5. Weibull modulus results

8.3.3. Introduction of defectsCeramic materials usually are subjected to mechanical failures because they are very sensitive to defects and flaws present into the microstructure. These defects are generated during the fabrication process and generally are: micro-pores, inclusions, micro-cracks and so on. One of the capabilities of the methodology developed consists in the fact that it is possible to introduce a dispersion of defects into the material. This is achieved by assigning different values of the stress level f that causes the opening of the interface.

This capability was first used to understand the role of the defect distribution chosen on the overall fracture properties of the material. Uniform and normal distributions were used to understand if, even in

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these cases, the results could be well interpolated by a Weibull cumulative probability function. Hence subsets of 50 simulations each were performed. Every subset is characterized by an assigned fracture behavior of the material, that can belong to one of the categories mentioned in paragraph 8.3 and by a specified statistical distribution of defects. Both for normal and for uniform distributions three sub-classes, characterized by the spread of the distribution, were defined.For the uniform distribution the spread, is defined as the ratio between one half of the length of the interval where the probability distribution is not null and the nominal stress, expressed in percent. It can be respectively: 10%, 20% or 30%.For the normal distribution the spread is the ratio of the 'three sigma' value for the distribution (that is three times the value of the standard deviation) over the nominal stress, expressed in percent. Even in this case it can be: 10%, 20% or 30%.Figure 8.9 shows an example of the three possible distributions for a fixed value of the nominal stress of the material and it makes clearer the reason for the two definitions of the spread used. In fact, fixing the nominal value of the critical stress f and the value of the spread, the total probability obtained by integrating the distribution function in the interval [1− f , 1 f ] is close to the unity for both the two distributions and therefore makes the results obtained using the same spread for the different distributions comparable.The results of the simulations were interpolated making use of different statistical functions (like normal, log-normal, beta, chi and Rayleigh functions), but Weibull distribution came out the best fitting one. Tests were carried using a tool embedded in the Matlab® environment that evaluates the quality of the fit. Figures 8.10 and 8.11 are an example of the good agreement between the numerical data and the Weibull distribution for two cases.

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Figure 8.9. Example of normal (blue) and uniform (red) stress distributions with different spreads. From top to bottom: 10%, 20% and 30% of the nominal stress.

Another aspect that can be underlined is that the values obtained for the Weibull parameters m and 0 are very similar both for the uniform and for the normal defect distributions, as it is possible to see from Table 8.7 and Table 8.8. The relative independence from the defect distribution of the final results was confirmed performing a set of simulations using a Weibull distribution for the generation of defects. A distribution with the same deviation of the Gaussian one with a 30% spread was generated. Comparing the results shown in Table 8.6 with the one displayed in Tables 8.8 and 8.7 it is possible to appreciate that the values of Weibull parameter reduced are very similar.Increasing the value of the spread of the distribution, the reduction of the Weibull modulus and of the threshold Weibull parameter, as expected, is measured for the material with the same resistance for grains and grain boundaries and for the material with grains weaker that grain boundaries.

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Figure 8.10. Failure cumulative plot for the material with weak grain boundaries with different values of uniform defects distribution

Figure 8.11. Failure cumulative plot for the material with the same resistance on grains and grain boundaries with different values of normal defects distribution

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The material with grain boundaries weaker than grains shows an unexpected different behavior, being scarcely influenced by the value of defects spread. The cause of this could be attributed to the fact that in this case the crack would be constrained to propagate only on a relatively small number of predefined paths, i.e. the grain boundaries. Let us imagine that the crack tip is on an interface whose opening value is below the nominal value of the grain boundary resistance and that the next interface to open has a relatively high opening stress. Since the crack is forced to continue the propagation on the grain boundary, being the grain much more resistant, the level of the force applied should be increased until the critical stress is not exceeded on the next edge. The presence of such an edge on the crack path causes the force applied to be increased to obtain a further propagation. The increase of the force necessary to cause the complete crack propagation can be therefore the reason of an increase of the nominal stress acted during the test and in turn of the value of 0 . This mechanisms are not supposed to be valid for the other two categories of materials, were the crack path is not constrained. In that cases the crack path will involve the bigger number as possible of weak interfaces. This could be a possible explanation why in those cases the value of 0 is always closer to the minimum stress given by the distribution, while in the other it is relatively close to the mean value.

Table 8.6. Results from the data reduction of Weibull distributed defects simulations

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Table 8.7: Results from the data reduction of uniform distributed defects simulations

Table 8.8: Results from the data reduction of normal distributed defects simulations

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Chapter 9

Fracture and shock assessment of MEMS accelerometers

9.1. Methodology adopted

An example of impact simulation for a micro accelerometer is presented in this Chapter. The results are obtained after a three level (here called global, device, detail levels) decoupled, simplified multi-scale simulation. This is necessary due to the fact that the dimensions of devices which can experience impacts due to accidental drop (e.g. centimeters for mobile phones) can be various order of magnitudes bigger then the dimensions of structural parts in MEMS which can break due to impact (fractions of micrometers). A numerical simulation to perform on a discretized model capable to describe the microstructural morphology of the material of the entire sensor, together with all other parts of the device would require billions of degrees of freedom and its CPU time would be in the order of weeks, if not months.The simplified decoupled simulation allows for a much faster total CPU time, but involves the assumption of some hypotheses that will be discussed in the following paragraphs.As an example the accidental drop of an inertial uniaxial accelerometer will be discussed in the following paragraphs. Since the sensor modeled and analyzed in this Chapter is property of STMicroelectronics, all dimensions that concern the device are intentionally omitted.

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9.2. Shock assessment of an uniaxial accelerometer

9.2.1. Global levelIn the upper-scale simulation the die of the inertial accelerometer is modeled. The word die is used referring to a system made up of (see Figure 9.1):

• the substrate; • the cap of the device;• the sensor; • the glass-frit ring that glues together the substrate and the cap.

As it is possible to see in bottom drawing of Figure 9.1, that is a section of the die, between the cap and the substrate a cavity is present. The MEMS accelerometer is placed inside the cavity, anchored to the substrate. The Finite Element model was used to evaluate the level of acceleration pulse that is transmitted to the sensor if the die is subjected to a free fall with an height of 150 cm. The orientation of the die with respect to the target surface is showed in Figure 9.2. The target is assumed to be perfectly rigid and the contact was modeled as frictionless.Since the ratio between the inertia of the sensor and that of the die assembly is very small, the dynamics of the whole device after the impact is marginally affected by the presence of the MEMS; for this reason only the die was modeled and then simulated. The model was discretized with 144298 linear tetrahedral elements and the problem was solved using the explicit solver present in the Abaqus Finite Element software.During the analysis, the displacement of the anchor points between the substrate and the device were stored. Figure 9.3 shows the displacement of the anchor points in the y-direction (with reference to reference frame of Figure 9.2).Further details of this simulation, as well as for the device level simulation, can be found in the work Ghisi A.et al., (2007).

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Figure 9.1. Layout of the die and Finite Element mesh

Figure 9.2. Die orientation during the free fall analysis

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Figure 9.3. Displacement of the anchor point in y-direction

9.2.2. Device levelThe results of the first simulation were used as input for the second one carried out on the whole polysilicon MEMS. The displacement time history u(t) computed in the first phase and shown in Figure 9.3, was used as input to the second analysis, by applying the displacement history to the anchor surface. The 3D model of the accelerometer is given in Figure 9.4. The model was discretized making use of 55888 8-noded brick elements. A time interval of 50 microseconds was considered and the Young’s modulus and the Poisson’s ratio of the MEMS were chosen equal to those of THELMATM polysilicon, i.e. 145000 MPa and 0.2, respectively. In order to take into account the holes in the suspended mass, that were not considered in the FE model, a reduced mass density

=1661 kg/m3 was chosen. The computed displacements on the section B-B (see Figure 9.5) at a distance 4.1 µm from the anchor in the left suspended beam , were used to prepare the input for the third level analysis. Figure 9.6 shows the vertical and horizontal displacement time history of the points in the selected section, respectively.The displacement in the out of plane direction was not plotted because its value is negligible. The orientation of the die before the impact was in fact chosen in order to cause a movement of the MEMS device in the plane parallel to the substrate, as illustrated in Figure 9.4.

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Figure 9.4. 3D views of the device. On the left a bottom view with highlighted the anchor between device and substrate; on the right a top view

Figure 9.5. Top view with a detail of the FE mesh nearby the anchor points.

Figure 9.6. Displacements on the section B-B

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This, in turn, makes possible the use of 2D models. In a general case, the shock produced by the fall would generate a full three dimensional stress state into the body.

9.2.3. Local levelIn order to verify the resistance of the device in correspondence of the connection of the suspension spring to the anchor, a third level simulation was performed on a model which can simulate crack initiation and propagation as described in Chapters 6 and 7. The tessellation of the anchor-beam detail analyzed is shown in Figure 9.7, together with the mesh used (7645 6-noded triangular elements). The orientation of each crystal was randomly obtained, the following elastic properties were assigned in the local reference system at each

crystal. The mass density =2330 kg/m3 of polysilicon was assigned to the model. The behavior of cohesive interfaces was governed by a linear softening law described in Chapter 6. The maximum allowable time step

associated with the fine time scale was on the order of 10-12 s, having the coarse time step a value 16 times bigger than the fine one.Two simulations were carried out using two different material interface models, both with the value of energy release rate equal to 7 N/m:

• one with the same strength properties on grains and grain boundaries (maximum admissible traction tmax=2.0 GPa on grains and grain boundary), referred to as Material I;

• one with grain boundaries weaker ( tmax=1.5 GPa ) than grains ( tmax=2.0 GPa ), referred to as Material II.

On the side marked in black in the left drawing of Figure 9.7 the displacement computed in the device level simulation is imposed.The other side is constrained by imposing a null displacement. The two simulations performed using different interface material parameters showed that the displacement imposed causes the complete fracture of the device. The complete fracture was obtained in about 13.5 µs. The crack evolution during the analysis time is displayed in Figure 9.8 and is very similar for the analysis carried out with Material I and Material II.

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Figure 9.7. Microstructual morphology and mesh of the detail

Figure 9.8. Crack Length vs. time for Material I (blue) and Material II (red)

Figure 9.9. Final crack paths for Material I (left) and Material II (right)

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From Figure 9.9 it is possible to notice that the crack path is the same for the two cases analyzed during the propagation in the first two grains and it is trans-granular. Once the crack length reaches almost one half of the length of the section, in the case analyzed using the material with same properties on grain and grain boundary, it continues the trans-granular propagation, while in the other case, it kinks on the grain boundary to terminate the propagation again trans-granular. In both the cases the two paths are very similar and even the time evolution of the crack length can almost be superimposed. As a closing remark, some critical comments on the procedure can be done. The global level analysis was performed disregarding the viscous effect of the air during the free fall of the die. This, together with the assumption of perfect rigidity of the target surface, leads to an overestimation of the shock induced on the MEMS device. The two simulations performed can be therefore seen as 'worst scenarios' and will give conservative results in terms of shock resistance.A more precise shock assessment should require a relatively high number of simulations performed on nominally identical specimens, in order to take into account the effect of the random orientation of the grains and of the different possible microstructural morphologies. The objective of this Chapter was just to explain the decoupled multi-scale procedure with reference to an example relatively simple in order to make possible the use of bi-dimensional models.

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Chapter 10

Conclusions

10.1. Achieved results

10.1.1. Experimental resultsThis part of the work was accomplished using the design technologies and production facilities of STMicroelectronics. Two different structures were designed and then measured in order to have a better understanding of the mechanical properties of two different structural layers fabricated using the ThELMATM process.The thin layer, called poly1, is a 0.7 µm thick polysilicon film. Previous works (see Cacchione et al., 2004) allowed for a mechanical characterization of this layer using a structure with a movement in a direction parallel to the plane of the wafer. The new structure tested was designed to understand if a load acting in a direction orthogonal to the plane of the wafer causes a different mechanical response of the material. This research was motivated by the fact that in the first case the mostly stressed parts of the structure correspond to the sidewalls that are the surfaces orthogonal to the wafer obtained with the etching process, while the latter structure described was designed in order to provoke a stress concentration mainly in the top and bottom surface of the film. Since these surfaces are not subjected to the etching process, they could present a different defect distribution and dispersion. This in turn can be the cause of a different fracture response of the material. From experimental results it was showed that indeed the surfaces present a different defectosity and as a consequence a different fracture resistance.

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The second structure presented was designed to load and break the epipoly, the thick layer produced with the ThELMATM process. The data obtained allowed for the determination of the Young's modulus that reveled to be consistent with results obtained in previous works carried by Corigliano and coworkers (2004). Weibull data reduction was performed and the results obtained showed that the material is very resistant and reliable. The experimental phase on this structure is not yet terminated and additional work is going on with the aim of a fatigue and fracture mechanics oriented characterization.

10.1.2. Numerical procedures and algorithmsA Voronoi tessellation procedure was implemented to construct a microstructural morphology similar to the one of polysilicon films. The procedure has three main parameters that characterize the numerical microstructure: the average grain size, that measure how fine the granulometry of the material is; the volume fraction of amorphous phase in the material (that in this work was always supposed to be null), that can take into account the presence of a second phase between different grains; the number of regularizations, that is zero if one wants to represent a material with distorted grains or a positive number if one wants to activate an iterative procedure that makes the shape of grains every time more regular.In Chapter 7 the algorithms for the time integration of the dynamic fracture or quasi-static fracture problem at the microscale were discussed. The problems connected with the microscale peculiarities were discussed and two strategies for the reduction of the computational cost of the simulations were proposed. The first consists in the adoption of an implicit integrator that exploits the linearity of the problem before the crack propagation and is therefore unconditionally stable. The second one consists in using in a way different from its original formulation a multi-domain decomposition method. By the application of the two techniques described it is possible to reduce almost of two orders of magnitude the computational cost of a simulation.

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10.1.3. Parametric simulationsIn Chapter 8 a parametric study of the mechanical properties of polysilicon is presented. The original contribution consists in the use of the numerical tools developed in order to perform 'virtual tests' to understand the role of microstructural parameters in the determination of elastic and fracture properties. The parameters investigated were: the average grain size and the crystalline privileged orientation for both elastic and fracture simulations; in addition for fracture simulations the size effect and the influence of a defect dispersion were taken into account.

10.1.4. Simplified approach for the shock assessmentThe development of efficient numerical algorithms for dynamic fracture simulations can be considered as a preliminary study in order to use numerical Finite element models for the simulation of impacts and shock loads on MEMS. In Chapter 9 an application of a simplified methodology is discussed. The example done shows that, assuming some simplifying hypotheses, it is possible, with reasonable computational times, to assess the shock resistance of the mostly stressed zones of the device.

10.2. Future prospects

10.2.1. Fatigue and fracture testingThe work described in Chapter 5 on the thick polysilicon test structure is going on within a collaboration project with a group of the Department of Electronic Engineering of Politecnico di Milano. The main topics to be tackled in the future are: the fatigue characterization of the material and the fracture mechanics testing. For this reason complex electronic layouts to control the experimental test were designed. The objective of this joint research could lead to the study of the influence of some environmental variables (humidity, temperature and pressure) on fatigue resistance, as well as more accurate determination of fracture properties of the material, that actually can be determined very approximatively.

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10.2.2. Numerical modelsOn the front of numerical simulations the development of fully three dimensional Finite element code appears necessary. Numerical geometrical models created with the tessellation procedures should be extended for this aim, as well as the routines that manage the changing topology of the system that in three dimensions are by far more complex. The domain decomposition algorithm could be applied in a more extensive way. Besides the use proposed in this work in order to increase the efficiency of the time integration, it could be used to run a complete fracture dynamics analysis of a device. The part supposed to undergo fracture would be integrated with a fine time step, while the remaining of the device would be integrated with a coarse time step. The computational cost of a complete simulation would be very high, but these problems can be solved optimizing or even parallelizing of the code running it on many CPUs.One of the most important factors that determine the quality and reliability of the simulation is the right choice of constitutive models for the materials. The choice of linear elastic models, confirmed by experimental results, appears appropriate for very brittle polycrystals as polysilicon. Two points that can be considered as crucial and that can be improved in future works, are: the definition of a fracture propagation criterion that takes into account the anisotropy of the material and its crystalline structure; the adoption of more complex and appropriate cohesive laws that can take into account rate effects, anisotropy effects and so on. The problem linked with this last points is that the use of complex material models is directly related to experiments. The definition of a model requires experiments for the control of the accuracy of the model proposed and then for the determination of materials parameters. As it was pointed out in Chapter 5, since the very small size of these devices and the difficulty to perform complex experiments tests, many of the informations required for the use of the cited models are still missing.

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